Matrix multiplication:
1) Find the product 𝐴 × 𝐵 for the matrices
2
𝐴=(
0
3 −1
),
−1 1
2
𝐵 = (1
3
−1
0)
1
2) Find the product 𝐴 × 𝐵 for the matrices
3 0
𝐴=(
),
2 1
2 3
𝐵=(
−2 4
1
)
0
3) Find the product 𝐴 × 𝐵 for the matrices
1 0 1
𝐴=(
),
0 4 2
−1 0
0
𝐵 = (1
−2 3)
2
0
1
4) Find the product 𝐴 × 𝐵 for the matrices
1
𝐴=(
2
0 −1
),
−1 1
1 0
𝐵 = ( 1 2)
−1 1
5) Find the product 𝐴 × 𝐵 for the matrices
2
𝐴 = (0
1
1 −1
−1 1 ) ,
1
0
2 −1
𝐵 = (1 0 )
3 1
6) Find the product 𝐴 × 𝐵 for the matrices
2
0
𝐴 = (−1 1 ) ,
1 −1
𝐵=(
1 1 −2
)
−2 4 0
7) Find the product 𝐴 × 𝐵 for the matrices
1 −2 1
𝐴=(
),
0 1 2
0
−1 1
𝐵 = (1
0
3)
−2 0
1
8) Find the product 𝐴 × 𝐵 for the matrices
2 3 −1
𝐴=(
),
0 −1 1
2 −1
𝐵 = (1 0 )
3 1
9) Find the product 𝐴 × 𝐵 for the matrices
3 0
𝐴=(
),
2 1
𝐵=(
2 3 1
)
−2 4 0
10) Find the product 𝐴 × 𝐵 for the matrices
1 0 1
𝐴=(
),
0 4 2
0 −1 2
𝐵 = (2 3
−1)
0 2
1
Determinants
1)
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1 2
|2 1
−1 0
2)
0
2|
3
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1 1
|−3 0
−1 3
3)
2
1|
0
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1
|−2
2
4)
0 −1
3 0|
1 −2
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1 2 −1
| 0 2 −3|
−1 1 3
5)
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1
|2
−2
6)
1 2
−1 0|
0 2
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
−1 1
|1 0
−2 2
7)
3
2|
0
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
0
|2
−1
1 2
−1 2|
0 3
8)
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
−1
|2
3
9)
0 −1
2 −3|
1 0
Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
−1 2
| 0 −1
2 −2
−1
2|
3
10) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
−1 0 2
|−2 1 3 |
3 2 −4
11) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1
3 0
|2
0 1|
−1 −1 2
12) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing them into triangular form
2
3
|1
0
−2 −1
−1
2|
0
13) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
−1 0 −1
|3
1
0|
2 −2 3
14) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1 −1 2
|−2 0 4|
2 −3 0
15) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
2
2
|−1 0
0 −1
−1
3|
−2
16) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1 0 −1
| 2 2 −3|
−3 1 0
17) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
1
|2
−2
1 2
−1 0|
0 2
18) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
0
3
| 2 −1
−1 0
1
−2|
2
19) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
2
|1
−3
2 3
−2 0|
0 2
20) Calculate the following determinant by three methods: by definition, by row- or
column-expansion method and by reducing it into a triangular form
−1
|1
2
2 3
0 2|
−2 0
Rank of matrices
1) Reduce the following matrix into a row-echelon form and identify its rank
1
2 −2
−2 −2 0
(
1 −1 2
0 −1 0
2
−1
)
1
2
2) Reduce the following matrix into a row-echelon form and identify its rank
2
1 −2
−1 0
1
(
2 −1 3
0
2 −5
3
4
)
1
2
3) Reduce the following matrix into a row-echelon form and identify its rank
0
1 −2 1
2 −1 0 1
(
)
−1 1 −1 2
1
0 −1 3
4) Reduce the following matrix into a row-echelon form and identify its rank
1
2
(
0
−1
2
1
1
2
1 −1
−2 0
)
−2 −1
1 −2
5) Reduce the following matrix into a row-echelon form and identify its rank
0 −1 −2
2 −2 0
(
−1 3
2
1
1
2
2
−1
)
0
−1
6) Reduce the following matrix into a row-echelon form and identify its rank
0
1
3 4
−2 3 −3 0
(
)
1 −1 3 2
−1 2
0 2
7) Reduce the following matrix into a row-echelon form and identify its rank
1
2 −1
2 −1 0
(
−2 0
2
−1 2
1
0
1
)
−1
−1
8) Reduce the following matrix into a row-echelon form and identify its rank
0
1 −2
−1 −2 1
(
0 −2 2
2
2
0
3
0
)
−1
−1
9) Reduce the following matrix into a row-echelon form and identify its rank
1
0
0
2
1 −2
(
−1 −1 1
2
1 −1
2
−1
)
1
1
10) Reduce the following matrix into a row-echelon form and identify its rank
2
1 −1
−1 −1 −1
(
1
0 −2
1
1
3
1
1
)
2
−1
Inverse matrix
1)
Identify if the following matrix is invertible or not. If invertible, find its inverse.
1
(2
0
2)
−2 −3
−1 4 )
2
0
Identify if the following matrix is invertible or not. If invertible, find its inverse.
2 1
0
(2 2
1)
0 −3 −1
3)
Identify if the following matrix is invertible or not. If invertible, find its inverse.
0 1
(2 −1
1 2
4)
Identify if the following matrix is invertible or not. If invertible, find its inverse.
1 3
(1 0
−1 2
5)
0
−2)
1
0
−2)
0
Identify if the following matrix is invertible or not. If invertible, find its inverse.
0
3 1
( 1 −1 1)
−2 2 0
6)
Identify if the following matrix is invertible or not. If invertible, find its inverse.
−1 1
(0 1
2 1
7)
Identify if the following matrix is invertible or not. If invertible, find its inverse.
1 −2
(2 1
0 2
8)
0
−2)
−1
−3
4)
0
Identify if the following matrix is invertible or not. If invertible, find its inverse.
−1 0 −3
( 2 −1 0 )
0 −2 −1
9)
Identify if the following matrix is invertible or not. If invertible, find its inverse.
0
1 −1
( 2 −1 −2)
−1 2
0
10) Identify if the following matrix is invertible or not. If invertible, find its inverse.
2 −1 0
( 0 −1 −2)
−1 −1 2
Vector systems
1)
Identify if the following vectors are linearly dependent or not
𝐴1 = (1 −1 3),
2)
1 3),
−1),
−1),
𝐴3 = (1 1 −2)
𝐴2 = (1 3 −2),
𝐴3 = (0
2 0)
𝐴2 = (1 −2
2),
𝐴3 = (−3
0 1)
𝐴2 = (1 −2 −2),
𝐴3 = (−1
−1 1)
𝐴2 = (0 −2 1),
𝐴3 = ( 1 1 0)
Identify if the following vectors are linearly dependent or not
𝐴1 = (0 1 2),
9)
𝐴2 = (0 −2 1),
Identify if the following vectors are linearly dependent or not
𝐴1 = (0 3
8)
𝐴3 = (−3 1 1)
Identify if the following vectors are linearly dependent or not
𝐴1 = (−2 1 3),
7)
𝐴2 = (1 0 2),
Identify if the following vectors are linearly dependent or not
𝐴1 = (2 −1 3),
6)
0 1)
Identify if the following vectors are linearly dependent or not
𝐴1 = (2 1 2),
5)
𝐴3 = (−3
Identify if the following vectors are linearly dependent or not
𝐴1 = (1 −3
4)
2),
Identify if the following vectors are linearly dependent or not
𝐴1 = (−2
3)
𝐴2 = (1 −2
𝐴2 = (2 3 −2),
𝐴3 = (0
2 0)
Identify if the following vectors are linearly dependent or not
𝐴1 = (−1 2 0),
𝐴2 = (−2 2
1),
𝐴3 = (1
0 −1)
10) Identify if the following vectors are linearly dependent or not
𝐴1 = (−2 1 3),
𝐴2 = (2 0
1),
𝐴3 = (0
1 1)
Systems of linear equations
1) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
2𝑦 + 3𝑧 = 3
𝑎) { 𝑥 + 𝑦 − 2𝑧 = 1
2𝑥 + 4𝑦 − 𝑧 = 5
𝑥−𝑦 =0
𝑏) { 𝑥 + 2𝑧 = 0
−𝑥 + 𝑦 − 𝑧 = 1
2) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
𝑥+𝑦+𝑧 =1
𝑎) { 𝑥 − 𝑦 + 𝑧 = 2
−𝑥 + 3𝑦 − 𝑧 = 3
𝑥−𝑦+𝑧 =0
𝑏) { 2𝑥 + 𝑦 = 0
−2𝑥 + 2𝑦 − 𝑧 = 3
3) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
2𝑥 + 2𝑦 − 𝑧 = 1
𝑎) { 𝑥 − 𝑧 = −1
−𝑥 − 2𝑦 = −2
2𝑦 + 3𝑧 = 3
𝑏) { 𝑥 − 2𝑦 − 𝑧 = 1
−2𝑥 + 𝑦 + 𝑧 = −3
4) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
−3𝑥 + 𝑦 + 𝑧 = 4
𝑎) {−2𝑥 − 2𝑦 + 𝑧 = 1
−𝑥 + 3𝑦 = 3
𝑥+𝑦−𝑧 =2
b) { 2𝑥 − 𝑦 = 0
−𝑥 + 𝑦 + 𝑧 = 2
5) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
𝑥 + 2𝑦 + 3𝑧 = 0
𝑎) {−𝑥 + 𝑦 + 2𝑧 = 1
2𝑥 + 𝑦 + 𝑧 = −2
2𝑥 + 𝑦 − 𝑧 = 1
𝑏) { 𝑥 − 𝑧 = −1
−𝑥 + 2𝑦 = −2
6) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
𝑥 + 𝑧 = −1
𝑥+𝑦 =1
𝑎) {
−𝑥 − 2𝑦 + 𝑧 = −3
2𝑥 + 𝑦 − 𝑧 = 2
𝑏) { 𝑥 − 𝑦 + 𝑧 = 1
−𝑥 + 𝑦 + 2𝑧 = 2
7) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
2𝑥 − 𝑦 − 𝑧 = 3
𝑎) { 𝑥 + 𝑦 − 2𝑧 = 1
−𝑥 + 2𝑦 − 𝑧 = −2
2𝑥 + 𝑦 − 𝑧 = 1
𝑏) { 𝑥 − 𝑧 = −1
−𝑥 − 𝑦 = −2
8) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
𝑥+𝑦+𝑧 =1
𝑎) { 𝑥 − 𝑦 + 𝑧 = 2
−𝑥 + 3𝑦 − 𝑧 = −3
𝑥−𝑦 =0
𝑏) { 𝑥 + 2𝑧 = 0
−𝑥 + 𝑦 − 𝑧 = 1
9) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
2𝑥 + 2𝑦 − 𝑧 = 0
𝑎) { −𝑥 + 𝑦 + 2𝑧 = 1
−𝑥 + 𝑦 − 3𝑧 = −3
𝑥+𝑦+𝑧 =1
𝑏) { −𝑥 + 𝑦 = 1
2𝑥 − 𝑦 − 𝑧 = 2
10) IIdentify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
−3𝑥 + 𝑦 + 𝑧 = 4
𝑎) {−2𝑥 − 2𝑦 + 𝑧 = 1
−𝑥 + 3𝑦 = 3
𝑥+𝑦−𝑧 =1
𝑏) {𝑥 − 2𝑦 + 3𝑧 = 3
2𝑥 − 𝑧 = 0
11) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
2𝑦 + 3𝑧 = 3
𝑎) { 𝑥 + 𝑦 − 2𝑧 = 1
2𝑥 + 4𝑦 − 𝑧 = 5
𝑥+𝑧=1
𝑏) { 2𝑦 + 𝑧 = 0
−𝑥 − 2𝑦 = 3
12) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
𝑥+𝑦+𝑧 =1
𝑎) { 𝑥 − 𝑦 + 𝑧 = 2
−𝑥 + 3𝑦 − 𝑧 = 3
𝑥−𝑦 =0
𝑏) { 2𝑥 + 𝑦 + 𝑧 = 1
−𝑥 + 𝑦 − 𝑧 = 2
13) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
method, matrix method and Cramer’s rule
2𝑥 + 2𝑦 − 𝑧 = 1
𝑎) { 𝑥 − 𝑧 = −1
−𝑥 − 2𝑦 = −2
2𝑥 − 𝑦 − 𝑧 = 2
𝑏) { 2𝑦 + 𝑧 = −1
−2𝑥 + 3𝑦 + 2𝑧 = 4
14) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
mtod, matrix method and Cramer’s rule
𝑥 + 2𝑦 + 3𝑧 = 0
𝑎) {−𝑥 + 𝑦 + 2𝑧 = 1
2𝑥 + 𝑦 + 𝑧 = −2
𝑥−𝑦 =0
𝑏) { 2𝑥 + 𝑦 + 𝑧 = 1
−𝑥 + 𝑦 + 2𝑧 = 2
15) Identify if the following systems of equations have no solution, unique solution or
infinitely many solutions. If there is a unique solution, then solve it by Gaussian
mtod, matrix method and Cramer’s rule
𝑥−𝑦−𝑧 =0
𝑎) {𝑥 + 2𝑦 + 𝑧 = 1
−3𝑦 − 2𝑧 = 1
𝑥+𝑦+𝑧 =2
𝑏) {2𝑥 − 𝑦 − 𝑧 = 1
𝑦 − 𝑧 = −1
Fundamental solutions
1)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
−2𝑥 − 𝑧 = 0
{ 𝑥 + 2𝑦 + 𝑧 = 0
−𝑥 + 2𝑦 + 2𝑧 = 0
2)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
2𝑦 + 3𝑧 = 0
{ 𝑥 + 𝑦 − 2𝑧 = 0
2𝑥 + 4𝑦 − 𝑧 = 0
3)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥+𝑦+𝑧 =0
{ 𝑥−𝑦+𝑧 =0
−𝑥 + 3𝑦 − 𝑧 = 0
4)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥 + 𝑦 + 2𝑧 + 𝑡 = 0
2𝑥 − 𝑦 − 𝑧 = 0
{
−𝑥 + 2𝑦 + 3𝑧 + 𝑡 = 0
3𝑥 + 𝑧 + 𝑡 = 0
5)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥−𝑦−𝑧 =0
𝑦−𝑧+𝑡 =0
{
𝑥 − 2𝑧 + 𝑡 = 0
𝑥 − 2𝑦 − 𝑡 = 0
6)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥−𝑦+𝑡 =0
𝑥+𝑦+𝑧 =0
{
2𝑦 + 𝑧 + 𝑡 = 0
−𝑥 − 3𝑦 − 2𝑧 + 𝑡 = 0
7)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑦+𝑧 =1
{𝑥 + 𝑦 − 𝑧 = 2
𝑥 + 2𝑦 = 3
8)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥 + 2𝑦 + 3𝑧 = 0
{ −𝑥 + 𝑦 − 𝑧 = 1
2𝑥 + 𝑦 + 4𝑧 = −1
9)
Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
−𝑥 + 𝑦 + 𝑧 + 𝑡 = 4
𝑥+𝑦 =2
{
𝑦+𝑧−𝑡 =0
𝑥 + 2𝑦 + 𝑧 − 𝑡 = 2
10) Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥 − 𝑦 − 2𝑡 = −1
𝑥 + 𝑦 − 2𝑧 + 2𝑡 = 1
{
𝑥−𝑧 =0
2𝑥 + 𝑦 − 3𝑧 + 2𝑡 = 1
11) Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥−𝑦−𝑧 =2
𝑦−𝑧+𝑡 =1
{
𝑥−𝑧+𝑡 =3
𝑥 − 2𝑦 − 𝑡 = 1
12) Show that the following system of linear equations have infinitely many
solutions. Then, find the fundamental solutions and basis vectors
𝑥+𝑦+𝑧 =2
−2𝑥 − 𝑧 + 𝑡 = 0
{
−𝑥 + 𝑦 + 𝑡 = 2
2𝑦 + 𝑧 + 𝑡 = 4
Linear Spaces (not necessary to solve
since in the exams these problems are not used)
1)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 + 𝑦 + 𝑧 = 1}
2)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |2𝑥 + 𝑦 = 0, 𝑧 + 1 = 0}
3)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 + 𝑧 = 2}
4)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 0, 𝑦 = 𝑧 2 }
5)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 = 𝑦𝑧}
6)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 2 + 𝑦 2 = 𝑧}
7)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑧 ≥ −2}
8)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 + 𝑧 < 0}
9)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 = 𝑦, 𝑦 = 𝑧 + 2}
10)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥, 𝑦, 𝑧 ≥ 0}
11)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 + 𝑦 + 𝑥 = −1}
12)
Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥, 𝑦, 𝑧 < 0}
13) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 + 𝑦 + 𝑧 < 0}
14) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 𝑦, 𝑧 = 𝑦 + 1}
15) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 < 0, 𝑦 = 𝑧 = 0}
16) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 = 1, 𝑦 = 2𝑧}
17) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑦 = 𝑥𝑧}
18) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 𝑦 2 + 𝑧 2 }
19) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 + 𝑦 + 𝑧 ≥ −2}
20) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 2𝑦, 𝑧 = 1}
21) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 = 3𝑦, 𝑦 = 2𝑧 + 1}
22) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 0, 𝑦 = 𝑧 < 0}
23) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ2 |𝑥 = 𝑦, 𝑥 = 𝑧 − 1}
24) Show that the following subset of ℝ3 is not a subspace:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥, 𝑦, 𝑧 > 1}
Basis and dimension of linear spaces (not necessary to
solve since in the exams these problems are not used)
1)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 + 𝑧 = 0}
2)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 0}
3)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 𝑦, 𝑧 = 0}
4)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 = 𝑧}
5)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |2𝑥 + 3𝑦 = 𝑧}
6)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 2𝑦, 𝑧 = 3𝑦}
7)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 + 𝑧 = 0, 𝑧 = 2𝑦}
8)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |2𝑥 − 𝑦 = 𝑧}
9)
Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 3𝑦 = 0, 𝑦 = 2𝑧}
10) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |2𝑥 − 𝑦 = 0}
11) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 3𝑧, 𝑦 = 2𝑧}
12) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 0, 𝑦 = −𝑧}
13) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 𝑦, 𝑧 = 0}
14) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 𝑦 = 0}
15) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |2𝑥 + 𝑦 = 𝑧}
16) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 3𝑦, 𝑦 = −𝑧}
17) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 = 0, 𝑧 = 0}
18) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 3𝑦, 𝑧 = −2𝑦}
19) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 = 0, 𝑧 = 2𝑦}
20) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 2𝑦 − 𝑧 = 0, 𝑧 = 𝑦}
21) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = 0}
22) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑧 = 0, 𝑦 = 0}
23) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 = −3𝑧, 𝑦 = 4𝑧}
24) Find a basis for the following subspaces, and conclude their dimensions:
𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 |𝑥 + 𝑦 + 2𝑧 = 0}
Scalar and vector producs
1) Calculate scalar and vector products of the vectors 𝑎 = (2, −1,3) and 𝑏 = (1,4,0).
2) Calculate scalar and vector products of the vectors 𝑎 = (3,0, −1) and 𝑏 = (2, −1,1)
3) Calculate scalar and vector products of the vectors 𝑎 = (−2,0,3) and 𝑏 = (1,3, −1)
4) Calculate scalar and vector products of the vectors 𝑎 = (1, −1,0) and 𝑏 = (1, −1,1)
5) Calculate scalar and vector products of the vectors 𝑎 = (2, −1,3) and 𝑏 = (0, −2,0)
6) Calculate scalar and vector products of the vectors 𝑎 = (0, −1,2) and 𝑏 = (1,4,1)
7) Calculate scalar and vector products of the vectors 𝑎 = (1, −2,3) and 𝑏 = (0,3,2)
8) Calculate scalar and vector products of the vectors 𝑎 = (3,0, −1) and 𝑏 = (−2,4,0)
9) Calculate scalar and vector products of the vectors 𝑎 = (2,2, −3) and 𝑏 = (1,4, −1)
10) Calculate scalar and vector products of the vectors 𝑎 = (3, −2,0) and 𝑏 = (−2,1,1)
11) Calculate scalar and vector products of the vectors 𝑎 = (1, −1,3) and 𝑏 = (2,4,0).
12) Calculate scalar and vector products of the vectors 𝑎 = (3,2, −1) and 𝑏 = (2, −2,1)
13) Calculate scalar and vector products of the vectors 𝑎 = (−2,1,3) and 𝑏 = (2,3, −1)
14) Calculate scalar and vector products of the vectors 𝑎 = (1, −1,1) and 𝑏 = (1, −1,2)
15) Calculate scalar and vector products of the vectors 𝑎 = (3, −1,3) and 𝑏 = (1, −2,0)
16) Calculate scalar and vector products of the vectors 𝑎 = (2, −1,2) and 𝑏 = (1,2,2)
17) Calculate scalar and vector products of the vectors 𝑎 = (−1, −2,3) and 𝑏 = (0,3,2)
18) Calculate scalar and vector products of the vectors 𝑎 = (2,0, −1) and 𝑏 = (−2,1,0)
19) Calculate scalar and vector products of the vectors 𝑎 = (2,1, −1) and 𝑏 = (2,2, −1)
20) Calculate scalar and vector products of the vectors 𝑎 = (0, −2,1) and 𝑏 = (−2,1,1)
Linear operators
1)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ, 𝜑(𝑥, 𝑦) = 𝑥 + 𝑦 + 1
2)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (𝑥 + 𝑦, 𝑦 + 2)
3)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ, 𝜑(𝑥, 𝑦) = 2𝑥 + 1
4)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ3 , 𝜑(𝑥, 𝑦) = (1, 𝑥 + 𝑦, 𝑥)
5)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (𝑥 − 𝑦, 1)
6)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ3 , 𝜑(𝑥, 𝑦) = (𝑥, 2,2𝑦)
7)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ, 𝜑(𝑥, 𝑦) = 2𝑥 + 𝑦 − 1
8)
Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ3 , 𝜑(𝑥, 𝑦) = (3𝑥, 2𝑦 + 1, 𝑥 + 𝑦)
9)
Show that the following map cannot be a linear operator
𝜑: ℝ3 → ℝ3 , 𝜑(𝑥, 𝑦, 𝑧) = (𝑥 + 𝑦, 𝑦 + 𝑧, 𝑥 + 𝑧 + 1)
10) Show that the following map cannot be a linear operator
𝜑: ℝ3 → ℝ3 , 𝜑(𝑥, 𝑦, 𝑧) = (𝑥 + 𝑦, 𝑦 − 𝑧, 𝑧 = 1)
11) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ, 𝜑(𝑥, 𝑦) = 𝑥 2 − 𝑦 2
12) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (2, 𝑥 + 𝑦)
13) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ, 𝜑(𝑥, 𝑦) = 3𝑥 − 5
14) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ3 , 𝜑(𝑥, 𝑦) = (𝑥 + 𝑦, 𝑦, 𝑥 − 1)
15) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (0, 𝑥 + 𝑦 − 1)
16) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ3 , 𝜑(𝑥, 𝑦) = (1,2𝑥, 3𝑦)
17) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ, 𝜑(𝑥, 𝑦) = 𝑥 2 + 𝑦 2
18) Show that the following map cannot be a linear operator
𝜑: ℝ2 → ℝ3 , 𝜑(𝑥, 𝑦) = (𝑥 2 , 𝑦, 0)
19) Show that the following map cannot be a linear operator
𝜑: ℝ3 → ℝ3 , 𝜑(𝑥, 𝑦, 𝑧) = (𝑥 + 𝑧, 𝑦 + 𝑧, 1)
20) Show that the following map cannot be a linear operator
𝜑: ℝ3 → ℝ3 , 𝜑(𝑥, 𝑦, 𝑧) = (𝑥 + 1, 𝑦 − 1, 𝑧 − 1)
Eigenvalues and eigenvectors
1) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (𝑥 + 𝑦, 3𝑦 − 𝑥)
2) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (𝑥 − 2𝑦, 3𝑦)
3) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (𝑥 + 3𝑦, 2𝑦 + 2𝑥)
4) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (−𝑥 + 4𝑦, 𝑥 + 2𝑦)
5) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (3𝑥 + 𝑦, 5𝑥 − 𝑦)
6) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (−𝑥, 𝑦 − 2𝑦)
7) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (−2𝑥 + 3𝑦, 𝑥)
8) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (−2𝑥 + 𝑦, 2𝑥 − 3𝑦)
9) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (𝑥 + 3𝑦, 2𝑦 + 2𝑥)
10) Describe the following linear operator as a matrix transformation. Then, find
its eigenvalues and the corresponding eigenvectors
𝜑: ℝ2 → ℝ2 , 𝜑(𝑥, 𝑦) = (−𝑥, 𝑦 − 2𝑦)
Quadratic forms
1)
Describe the following quadratic form in a matrix form, and find its rank.
𝑞(𝑥1 , 𝑥2 ) = 𝑥12 + 6𝑥1 𝑥2 − 2𝑥22
2)
Describe the following quadratic form in a matrix form, and find its rank.
𝑞(𝑥1 , 𝑥2 ) = 2𝑥12 − 4𝑥1 𝑥2 + 3𝑥22
3)
Describe the following quadratic form in a matrix form, and find its rank.
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = −𝑥12 + 2𝑥1 𝑥2 + 𝑥22 − 4𝑥2 𝑥3 − 2𝑥22
4)
Describe the following quadratic form in a matrix form, and find its rank.
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = 2𝑥12 + 2𝑥1 𝑥3 + 3𝑥22 − 𝑥22
5)
Describe the following quadratic form in a matrix form, and find its rank.
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = 2𝑥1 𝑥2 + 4𝑥1 𝑥3 − 2𝑥2 𝑥3
6)
Describe the following quadratic form in a matrix form, and find its rank.
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = 𝑥12 − 2𝑥1 𝑥2 + 6𝑥1 𝑥3 − 𝑥22 + 4𝑥2 𝑥3
7)
Check if the following quadratic form is positively defined or not by calculating
main minors of is matrix
𝑞(𝑥1 , 𝑥2 ) = 𝑥12 − 6𝑥1 𝑥2 + 10𝑥22
8)
Check if the following quadratic form is positively defined or not by calculating
main minors of is matrix
𝑞(𝑥1 , 𝑥2 ) = 𝑥12 + 4𝑥1 𝑥2 + 2𝑥22
9)
Check if the following quadratic form is positively defined or not by calculating
main minors of is matrix
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = 3𝑥12 − 2𝑥1 𝑥2 + 𝑥22 − 4𝑥1 𝑥3 + 4𝑥22
10) Check if the following quadratic form is positively defined or not by calculating
main minors of is matrix
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = 2𝑥12 + 4𝑥1 𝑥2 − 𝑥22 − 4𝑥2 𝑥3 + 𝑥22
11) Check if the following quadratic form is positively defined or not by calculating
main minors of is matrix
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = 2𝑥12 − 2𝑥1 𝑥2 − 2𝑥1 𝑥3 + 𝑥22 + 2𝑥22
12) Check if the following quadratic form is positively defined or not by calculating
main minors of is matrix
𝑞(𝑥1 , 𝑥2 , 𝑥3 ) = 𝑥12 + 𝑥22 − 4𝑥1 𝑥2 + 𝑥32
Representation of curves
1)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
3𝑥 2 + 2𝑦 2 − 6𝑥 + 8𝑦 + 5 = 0
2)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
2𝑥 2 + 4𝑥𝑦 + 2𝑦 2 − 2𝑥 + 2 = 0
3) Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
𝑥 2 + 2𝑥𝑦 + 𝑦 2 + 𝑥 + 2𝑦 + 1 = 0
4)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
2𝑥 2 − 𝑥𝑦 − 𝑦 2 + 𝑥 − 4𝑦 − 3 = 0
5)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
𝑥 2 − 2𝑥𝑦 + 𝑦 2 + 3𝑥 − 𝑦 + 2 = 0
6)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
2𝑥 2 − 4𝑥𝑦 + 4𝑦 2 − 2𝑥 + 1 = 0
7)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
𝑥 2 − 2𝑥𝑦 + 3𝑦 2 − 4𝑦 + 1 = 0
8)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
𝑥 2 + 2𝑥𝑦 + 2𝑦 2 + 2𝑥 − 2𝑦 + 5 = 0
9)
Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
2𝑥 2 + 4𝑥𝑦 + 2𝑦 2 − 2𝑥 + 2 = 0
10) Classify the following second order equation (there can be 6 cases depending on
Δ and 𝛿):
2𝑥 2 − 4𝑥𝑦 + 4𝑦 2 − 2𝑥 + 1 = 0
Lines and planes in the space
1)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (1, −2), 𝐵 = (0,3)
2)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (2,0), 𝐵 = (−1,2)
3)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (2, −3), 𝐵 = (1,2)
4)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (1,0), 𝐵 = (3,4)
5)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (2,0), 𝐵 = (1,3)
6)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (−2,2), 𝐵 = (3,1)
7)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (0, −3), 𝐵 = (−1, −2)
8)
Find the general equation of the line passing through the given two points 𝐴 and 𝐵,
and sketch its graphs:
𝐴 = (2,2), 𝐵 = (−1,3)
9)
Find the angle between the given lines
2𝑥 + 3𝑦 − 1 = 0,
−𝑥 + 2𝑦 + 2 = 0
10) Find the angle between the given lines
−𝑥 + 2𝑦 − 1 = 0,
2𝑥 − 𝑦 + 2 = 0
11) Find the angle between the given lines
−2𝑥 − 2𝑦 + 3 = 0,
−𝑥 + 1 = 0
12) Find the angle between the given lines
𝑦 − 1 = 0,
−𝑥 + 𝑦 + 4 = 0
13) Find the angle between the given lines
𝑦 = 2𝑥 − 1,
𝑦 = −𝑥 + 2
14) Find the angle between the given lines
1
𝑦 = 3𝑥 − 1, 𝑦 = − 𝑥
3
15) Find the angle between the given lines
𝑦 = 3𝑥 − 1, 𝑦 = 3𝑥 + 2
16) Find the angle between the given lines
𝑦 = 𝑥 − 1, 𝑦 = −2𝑥
17) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (2, −3), 𝑣̅ = (1,2)
18) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (−1,4), 𝑣̅ = (2, −1)
19) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (0,0), 𝑣̅ = (1,1)
20) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (−1,0), 𝑣̅ = (3, −2)
21) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (−2,3), 𝑣̅ = (2,2)
22) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (0,3), 𝑣̅ = (3,1)
23) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (2,2), 𝑣̅ = (−1, −1)
24) Find the equations of lines passing through the point 𝐴 and parallel to the vector 𝑣̅ :
𝐴 = (−1,1), 𝑣̅ = (2, −1)
25) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (1, −2,0),
𝐵 = (2,1,1),
𝐶 = (0,0, −1)
26) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (1,1,0),
𝐵 = (−2, −1,3),
𝐶 = (1, −1,2)
27) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (0,0,0),
𝐵 = (1,2,0),
𝐶 = (1,1, −1𝑠)
28) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (−1, −1,0),
𝐵 = (1,0, −3),
𝐶 = (0, −2,0)
29) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (2, −2,0),
𝐵 = (2,1,1),
𝐶 = (1,0, −1)
30) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (−1,1,0),
𝐵 = (0, −1,3),
𝐶 = (0, −1,2)
31) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (1,0,0),
𝐵 = (1,1,0),
𝐶 = (1, −1,2)
32) Find the equation of planes passing through given three points 𝐴, 𝐵, and 𝐶:
𝐴 = (2,0,3),
𝐵 = (0, −2, −1),
𝐶 = (1,1,0)
Linear programming
1. Solve the linear programming using graphical method:
Maximize 𝑝 = 𝑥 + 𝑦
subject to:
𝑥 + 𝑦 ≤ 10
2𝑥 + 𝑦 ≤ 16
−𝑥 + 𝑦 ≤ 3
𝑥 ≥ 0, 𝑦 ≥ 0
2. Solve the linear programming using graphical method:
Minimize 𝑝 = 𝑥 + 3𝑦
subject to:
𝑥 + 2𝑦 ≥ 10
𝑥−𝑦 ≤3
𝑥 ≥ 0, 𝑦 ≥ 0
3. Solve the linear programming using graphical method:
Maximize 𝑝 = 8𝑥 + 𝑦
subject to:
𝑥 + 𝑦 ≤ 40
2𝑥 + 𝑦 ≤ 60
𝑥 ≥ 0, 𝑦 ≥ 0
4. Solve the linear programming using graphical method:
Maximize 𝑝 = 3𝑥 + 2𝑦
subject to:
2𝑥 + 𝑦 ≤ 10
𝑥+𝑦 ≤8
𝑥≤4
𝑥 ≥ 0, 𝑦 ≥ 0
5. Solve the linear programming using graphical method:
Maximize 𝑝 = 𝑥 + 2𝑦
subject to:
2𝑥 + 𝑦 ≤ 8
𝑥+𝑦 ≤5
𝑥 ≥ 0, 𝑦 ≥ 0
6. Solve the linear programming using graphical method:
Maximize 𝑝 = 6𝑥 − 9𝑦
subject to:
2𝑥 − 3𝑦 ≤ 6
𝑥 + 𝑦 ≤ 20
𝑥 ≥ 0, 𝑦 ≥ 0
7. Solve the linear programming using graphical method:
Maximize 𝑝 = 3𝑥 + 𝑦
subject to:
2𝑥 − 𝑦 ≥ 0
−𝑥 + 2𝑦 ≤ 3
𝑦≤3
𝑥 ≥ 0, 𝑦 ≥ 0
8. Solve the linear programming using graphical method:
Minimize 𝑝 = −3𝑥 − 𝑦
subject to:
4𝑥 − 𝑦 ≥ 0
2𝑥 − 𝑦 ≤ 0
𝑥+𝑦 ≤3
𝑥 ≥ 0, 𝑦 ≥ 0
9. Solve the linear programming using graphical method:
Minimize 𝑝 = 5𝑥 − 𝑦
subject to:
2𝑥 − 3𝑦 ≤ 0,
−5𝑥 + 9𝑦 ≤ 45,
𝑥 − 2𝑦 ≤ 4,
𝑥 ≥ 0, 𝑦 ≥ 0
10.Solve the linear programming using graphical method:
Maximize 𝑝 = 3𝑥 + 6𝑦
subject to:
−4𝑥 + 𝑦 ≥ 0
𝑥 − 𝑦 ≥ −3
2𝑥 − 3𝑦 ≤ 6
𝑥 ≥ 0, 𝑦 ≥ 0
Simplex method
11.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 + 4𝑥2 + 𝑥3
subject to:
−𝑥1 + 2𝑥2 + 𝑥3 ≤ 4,
3𝑥1 + 𝑥2 + 2𝑥3 ≤ 9,
2𝑥1 + 3𝑥2 + 𝑥3 ≥ 6,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
12.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 − 𝑥2 + 𝑥3
subject to:
4𝑥1 + 2𝑥2 + 𝑥3 ≥ 6,
−𝑥1 + 𝑥2 + 𝑥3 ≤ 1,
𝑥1 − 𝑥2 + 4𝑥3 ≤ 24,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
13.Solve the linear programming using Simplex method:
Maximize 𝑝 = 5𝑥1 + 2𝑥2 + 𝑥3
subject to:
𝑥1 + 𝑥2 + 𝑥3 ≥ 3,
𝑥1 + 2𝑥2 + 2𝑥3 ≤ 4,
3𝑥1 + 4𝑥2 + 2𝑥3 ≤ 12,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
14.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 − 8𝑥2 − 3𝑥3
subject to:
3𝑥1 + 𝑥2 + 2𝑥3 ≥ 6,
𝑥1 + 𝑥2 + 𝑥3 ≤ 4,
𝑥1 − 3𝑥2 + 𝑥3 ≤ −4,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
15.Solve the linear programming using Simplex method:
Maximize 𝑝 = −𝑥1 − 3𝑥2 − 𝑥3
subject to:
3𝑥1 + 𝑥2 + 𝑥3 ≥ 6,
𝑥1 + 3𝑥2 + 𝑥3 ≤ 10,
𝑥1 − 3𝑥2 + 𝑥3 ≤ −2,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
16.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 + 4𝑥2 + 3𝑥3
subject to:
𝑥1 − 3𝑥2 + 2𝑥3 ≤ 3,
2𝑥1 + 4𝑥2 + 𝑥3 ≤ 18,
−𝑥1 + 𝑥2 + 3𝑥3 ≥ 10,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
17.Solve the linear programming using Simplex method:
Maximize 𝑝 = −4𝑥1 − 3𝑥2 − 2𝑥3
subject to:
4𝑥1 + 𝑥2 + 2𝑥3 ≥ 8,
2𝑥1 + 𝑥2 − 𝑥3 ≤ 6,
𝑥1 − 3𝑥2 − 𝑥3 ≥ −4,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
18.Solve the linear programming using Simplex method:
Maximize 𝑝 = 4𝑥1 + 𝑥2 + 3𝑥3
subject to:
4𝑥1 − 𝑥2 − 2𝑥3 ≤ 3,
𝑥1 + 3𝑥2 + 𝑥3 ≥ 4,
3𝑥1 − 𝑥2 + 𝑥3 ≤ 12,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
19.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 − 3𝑥2 − 2𝑥3
subject to:
3𝑥1 + 𝑥2 − 2𝑥3 ≥ 13,
𝑥1 − 3𝑥2 + 𝑥3 ≤ 1,
𝑥1 + 2𝑥2 + 3𝑥3 ≤ 11,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
20.Solve the linear programming using Simplex method:
Maximize 𝑝 = 3𝑥1 + 2𝑥2 + 3𝑥3
subject to:
−𝑥1 − 𝑥2 + 2𝑥3 ≤ −2,
−𝑥1 + 2𝑥2 + 𝑥3 ≤ 4,
𝑥1 + 2𝑥3 ≤ 2,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
21.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 + 2𝑥2 + 𝑥3
subject to:
−𝑥1 − 𝑥2 + 𝑥3 ≤ −1,
𝑥1 + 𝑥2 + 𝑥3 ≤ 3,
𝑥1 + 𝑥3 ≤ 1,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
22.Solve the linear programming using Simplex method:
Maximize 𝑝 = 2𝑥1 + 𝑥2 + 2𝑥3
subject to:
𝑥1 + 2𝑥2 − 𝑥3 ≥ 2,
−2𝑥1 + 𝑥2 + 2𝑥3 ≤ 2,
−2𝑥1 − 𝑥2 + 2𝑥3 ≥ −6,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
23.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 + 4𝑥2 + 𝑥3
subject to:
−𝑥1 + 2𝑥2 + 𝑥3 ≤ 4,
3𝑥1 + 𝑥2 + 2𝑥3 ≤ 9,
2𝑥1 + 3𝑥2 + 𝑥3 ≥ 6,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
24.Solve the linear programming using Simplex method:
Maximize 𝑝 = 2𝑥1 + 𝑥2 − 𝑥3
subject to:
2𝑥1 + 𝑥2 − 𝑥3 ≥ 5,
𝑥1 + 2𝑥2 + 𝑥3 ≤ 7,
𝑥1 − 𝑥2 + 2𝑥3 ≤ 1,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
25.Solve the linear programming using Simplex method:
Maximize 𝑝 = 𝑥1 − 3𝑥2 − 2𝑥3
subject to:
3𝑥1 + 𝑥2 − 2𝑥3 ≥ 13,
𝑥1 − 3𝑥2 + 𝑥3 ≤ 1,
𝑥1 + 2𝑥2 + 3𝑥3 ≤ 11,
𝑥1 , 𝑥2 , 𝑥3 ≥ 0
Transport problem
1) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
4
3
2
5
𝐷2
2
1
4
3
𝐷2
0
3
1
4
𝐷4
1
2
0
3
150
250
75
225
200
100
300
300
2) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
0
1
2
3
𝐷2
2
1
2
3
𝐷2
0
2
1
4
𝐷4
1
0
0
3
100
500
100
100
150
150
200
300
3) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
1
3
3
5
𝐷2
2
2
2
3
𝐷2
0
4
1
0
𝐷4
3
2
0
1
200
400
400
100
800
100
150
50
4) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
2
1
2
3
𝐷2
2
1
4
5
𝐷2
2
5
1
4
𝐷4
1
2
0
0
500
250
250
200
200
400
350
250
5) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
1
2
3
1
𝐷2
2
2
5
0
𝐷2
3
4
5
4
𝐷4
4
1
0
2
200
200
150
50
150
50
250
150
6) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
2
3
1
0
𝐷2
5
2
4
3
𝐷2
4
3
2
4
𝐷4
1
2
0
2
100
300
400
300
300
150
250
400
7) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
2
4
2
2
𝐷2
2
0
2
3
𝐷2
4
3
1
0
𝐷4
2
2
1
3
300
300
400
400
550
450
250
150
8) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
2
1
2
4
𝐷2
3
0
4
4
𝐷2
1
3
2
4
𝐷4
1
4
0
4
800
100
200
300
300
400
150
650
9) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
1
4
1
3
𝐷2
2
5
1
3
𝐷2
0
3
5
0
𝐷4
2
2
3
2
175
175
250
400
150
150
250
450
10) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
1
2
3
2
𝐷2
3
2
4
0
𝐷2
1
3
2
5
𝐷4
5
4
0
4
300
300
200
200
300
200
100
400
11) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
2
4
5
3
𝐷2
2
0
1
1
𝐷2
0
4
5
0
𝐷4
1
2
3
0
100
50
50
200
150
50
100
100
12) Check if the following transport problem is balanced or not. Solve it
using Northwest and minimum cost methods. Compare the results.
𝑆1
𝑆2
𝑆3
𝑆4
𝐷1
2
2
3
3
𝐷2
0
2
4
2
𝐷2
1
1
2
1
𝐷4
5
4
2
4
200
300
400
300
150
250
350
450