1.
2.
3.
4.
5.
6.
7.
8.
Check whether the vectors V1= (1, 2, 3) V2= (3, 1, 2) V3= (2, 0, 1) are linearly independent or not.
Explain Gauss Elimination method for solving non homogenous system of linear equations.
Check whether the given function below is linear transformation or not.
(𝑥1)^2
𝑥1
T ={ [
]}=[
]
𝑥2
0
Check whether the following orthogonal set obeys the Pythagoras theorem.
u = (1, 8, 1), v = (3, 2, 1).
Explain about Diagonalization of vectors.
Show that the given matrix is Hermitian matrix.
0
2+𝑖
A= [
]
2 − 𝑖 −2
Explain linear dependent and independent vectors in terms of rank of a system.
1 2 1
Given that matrix A= [
], Find the rank of AAT.
0 1 1
9.
Check whether S = {(1, 2, 0, 1) (0, 0, 1, 5) (1, 1, 2, 1) (1, 2, 1, 6)} is a basis of R4.
10.
11.
Explain three types of algebraic structure and its properties
Find the consistent of the system of equation using rank.
2x - 2z = 6
y+z=1
2x + y – z = 7
3y + 3z = 0
Find basis and dimension for row space of the matrix.
1
2 0 2
−2
5 1 −1]
A=[
0 −3 3 4
3
6 0 −7
12.
13.
Find the change of basis of a given matrix from S1 to S2 and S2 to S1.
S1= {u1 = (1, -2), u2= (3, -4)}, S2 = {v1= (1, 3), v2= (3, 8)}
14.
1 0 2
Find the inverse of matrix A = [2 −1 3]
4 1 8
15.
Show that given vectors are orthogonal and find its orthonormal basis using Gram
Schmidt orthonormalization.
1
6
V1=[1] , V2 =[4]
1
5
If u = (1, 2, 3), v = (2, -3, 5) then find the inner product of <u.v>, Norm of ||u+v|| and
D(u,v).
−1 2 1
Given that A = [ 0 1 −1], Determine the eigen values and their algebraic
0 2 −2
multiplicities.
1 2
Given that matrix A = [
], Check whether the matrix A is diagonalizable or not.
0 1
1 2
Find the pseudo inverse of matrix A = [−1 3]
2 4
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
1 1
Find the SVD of the matrix A= [ 0 1]
−1 1
Find the basis and dimension of the vectors space generated by S = {(1,2) (1,1) (3,1)}.
Express the vector V= (1, -2, 5) in 3D vector space on a linear combination of the vector v1 = (1,1,1) v2
= (1, 2, 3) v3 = (2, -1, 1).
Explain the system of homogenous linear equation.
−2 −1
Find the inverse of matrix A = [
].
3
3
Explain Inner product, Norm and Distance in vector space
Show that given vectors are orthogonal and orthonormal basis.
3/√11
−1/√6
−1/√66
V1 = [1/√11] V2 = [ 2/√6 ]
V3 = [−4/√66]
1/√11
1/√6
7/√66
2 1 0
Let A = [0 2 0], Find the eigen values and their algebraic multiplicities.
0 0 1
Find the rank of AAT.
3 2 0
A= [
].
1 1 1
29.
Check whether the vectors V1= (2,1,3) V2= (5,0,3) V3= (3, -1,0) are linearly
independent or not.
30.
Explain the Algebraic system and its general properties
31.
Find the solution of the given linear system using Gauss elimination method.
x + 4y – z = -5
x + y - 6z = -12
3x – y – z = 4
Check whether the given linear system is Trivial or not.
x + 3y + 2z = 0
2x – y + 3z = 0
3x - 5y + 4z = 0
x + 17y + 4z = 0
Find all the fundamental subspace of the matrix given below.
1 −2 −1 3 2
A = [ 2 −2 −3 6 1 ]
−1 −4 4 −3 7
32.
33.
34.
35.
36.
37.
38.
Find the change of basis of a given matrix from S1 to S2 and S2 to S1.
S1= {u1 = (1,2), u2= (1,3)}, S2 = {v1= (3,1), v2= (0,1)}
find orthonormal basis of given vectors using Gram Schmidt orthonormalization.
3
2
V1= ( ) , V2= ( )
1
2
Check whether the following orthogonal set obeys the Pythagoras theorem.
u = (1, 2, -3, 4), v = (3, 4, 1, -2), w = (3, -2, 1, 1)
1 2
Diagonalize the matrix A = [
]
2 1
Check whether the given matrix is Hermitian or not.
0 𝑖
A=[
]
𝑖 0
39.
40.
1 −1
Find the SVD of the matrix A= [−2 2 ]
2 −2
Find the least square solution to the matrix equation using pseudo inverse method.
2 −2
−1
[−2 2 ] X = [ 7 ]
5
3
−26