Math 237
Carter
Study Guide Final
Summer 2013
I’ll build the final exam from this file. It is a compilation of several old exams and quizzes.
Definitions
1. The span of the set {~v1 , ~v2 , . . . , ~vk } ⊂ V is the set
{~v ∈ V : v =
k
X
αi~vk for some αi ∈ R}
i=1
2. If the homogeneous equation
x1~v1 + x2~v2 + . . . xk~vk = 0
has only the trivial solution x1 = x2 = . . . = xk = 0, then the set {~v1 , ~v2 , . . . , ~vk } ⊂ V
is said to be linearly independent.
3. The row space of a matrix A ∈ M (m × n) is the span of the set of rows. That is if


A1


 A2 


A =  . ,
.
 . 


m
A
P
j
then the row space is the set Row(A) = { m
j=1 αj A : αj ∈ R ∀ j = 1, . . . m}.
4. The column space of a matrix A ∈ M (m, n) is the span of the set of columns. That
is if
A = [A1 , A2 , . . . , An ] ,
P
then the column space is the set Col(A) = { m
i=1 αi Ai : αi ∈ R ∀j = 1, . . . n}.
5. The kernel (null space) of a matrix A ∈ M (m, n) is
~ : AX
~ = ~0) }.
ker(A) = null(A){X
6. A subspace W of a vector space V is a set of vectors that is non-empty and closed
under linear combinations.
W / V ⇔ ~0 ∈ W
whenever α, β ∈ R and w~1 , w~2 ∈ W .
1
& αw~1 + β w~2 ∈ W
7. A basis for a vector space V is a set {~v1 , ~v2 , . . . , ~vk } ⊂ V such that
(a) {~v1 , ~v2 , . . . , ~vk } spans V in the sense that any vector ~v ∈ V , can be written as
P
~v = ki=1 αi~vk for some scalars αi ∈ R, and
(b) the set {~v1 , ~v2 , . . . , ~vk } ⊂ Rn is linearly independent (see above).
Pk
Thus the representation ~v =
i=1
αi~vk is unique.
8. If the vector space V has a basis {~v1 , ~v2 , . . . , ~vk }, then the dimension of V is k, that
is, the number of elements in a linearly independent set that spans V .
F
9. A linear map U ←− V is a function such that
F (α~v + β w)
~ = αF (~v ) + βF (w)
~
for all α, β ∈ R and for all ~v , w
~ ∈V.
F
10. An eigen-vector for a linear operator Rn ←− Rn is a vector ~v ∈ Rn such that there
is a λ ∈ R with
A~v = λ~v .
The number λ is said to be the associated eigen-value.
Important Theorems
F
1. Let U ←− V denote a linear map from an n-dimensional vector space V to an mdimensional vector space U and let A ∈ M (n, m) denote any matrix representing F .
The rank-nullity theorem states
rank(F ) + dim(ker(F )) = n.
In particular, if n < m, then F cannot be surjective (onto); and if m < n, then
F cannot be injective. Many of the equivalences below follow from the rank-nullity
theorem.
2
2. The following statements are equivalent:
F
(a) Rn ←− Rn is non-singular.
(b) Any matrix A that represents F is non-singular.
(c) For any matrix A ∈ M (n, n) the equation AX = B has a solution for any B ∈
M (n, 1).
(d) For any matrix A ∈ M (n, n) the equation AX = B has a unique solution for any
B ∈ M (n, 1).
(e) For any matrix A ∈ M (n, n) the equation AX = ~0 has a unique solution.
(f)
det (A) 6= 0
(g) The rows of A form a linearly independent set.
(h) The columns of A form a linearly independent set.
(i)
Row(A) = M (1, n).
(j)
Col(A) = M (n, 1).
(k)
dim(Row(A)) = n.
(l)
dim(Col(A)) = n.
(m) The rows of A span M (1, n).
(n) The columns of A span M (n, 1).
(o) The linear operator F is injective (one-to-one).
(p) The linear operator F is surjective (onto).
(q)
ker(A) = {~0}.
(r) The rank of A is n.
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Computational Exercises
1. Give a parametric equation for the line perpendicular to the plane 2x − 3y + 7z = 4
that contains the point P (1, −5, 7).
2. Find the equation of the plane that is parallel to 4x + 3y − 2z = 11 and that contains
the point Q(2, −1, 3).
3. Express the solutions set to the following equations as a point in space plus the sum
of two parameters times directions as in the example below. Use y and z as your free
parameters.
(a)
3x + 4y + 8z = 24
(b)
2x − 5y − 6z = 30
(c)
x+y+z =1
(d)
5x + 4y − 2z = 20
Determine a vector that is perpendicular to the plane defined by the equation.
4. Solve the equation
a1 x 1 + a2 x 2 + · · · + an x n = b
for x1 . Give the intersections between this plane and the coordinate planes xj = 0.
Under what circumstances will there be no such intersection?
5. Compute the matrix product of the following matrices.
(a)
"
1 −5/2
0
# "
·
1
2 −5 30
0
1
0
2 −5
#
#
(b)
"
1/2 0
0
1
# "
·
0
1
(c)
"
1/3 −4/3
0
1
4
# "
·
3 4 24
0 1
0
#
(d)
"
1/3 0
0
# "
·
1
1 −4
0
#
1
6. Solve the system of equations:
+ 2y + 4z − 5t = 3
x
3x −
y
+ 5z + 2t = 4
5x − 4y + 6z + 9t = 2
7. Write the matrix
"
B=
3 −6
2
#
4
as a product of elementary matrices. Recall that a matrix is an elementary matrix
if it is obtained from the identity matrix by means of an elementary row operation.
8. Express the matrix
"
M=
4 7
#
7 9
as a linear combination of the matrices A, B, and C where
"
#
"
#
"
#
1 1
1 2
1 1
A=
, B=
, and C =
.
1 1
3 4
4 5
9. Find the dimension and a basis for the solution space W for the homogeneous system
of equations:
x
+
y
+ 2z = 0
2x + 3y + 3z = 0
x
+ 3y + 5z = 0
F
10. Let R3 ←−R4 denote the linear mapping that is given by

x



x − y + z + t
 
 y  




F = x
+ 2z − t 
.
 z 
x + y + 3z − 3t
t
Find a basis for the image of F , and find a basis for the null space (kernel) of F .
5
11. Suppose that the matrix
"
A=
2
3
#
4 −1
F
represents a linear operator R2 ←− R2 relative to the usual basis
(" # " #)
1
0
B=
,
0
1
for R2 . Find the matrix B that represents A with respect to the basis
(" # " #)
1
2
S=
,
.
3
5
12. Let A be a given (4×3)-matrix. Explain in complete sentences why there is an equation
AX = B which has no solution.
13. Determine the equation of the plane whose solution space is the span of the set of
vectors {[1, 1, 2]t , [1, 2, 0]t }.
14. Show that the set





a
b
c







U =  0 d e  : a, b, c, d, e, f ∈ R





 0 0 f
of upper triangular (3 × 3)-matrices is a subspace of the space M (3, 3). What is the
dimension of U?
15. Determine bases for the column space, row space, and null space of the matrix


1 2 −1 3


.
A=
2
2
−4
4


1 3 0 4
"
a b
#
denote a general (2 × 2)-matrix for which a 6= 0 and ad − bc 6= 0.
c d
Show, by using row reduction, the computation of the inverse matrix of A.
16. Let A =
6
17. For the following matrices compute the row space, column space, and null space.


1 0 4


 2 0 7 


(a) 

 1 2 1 
1 2 2
"
#
0 1 4 6 10
(b)
0 0 1 1 1


1
0 0
0



 0 −1 0
0

(c) 
 0
0 0 −1 


0
0 1
0


1 1 −1 1 1



(d) 
−1
1
−1
1
1


0 0
0 0 1
18. Compute the inverses of the following matrices, A, and solve the equation AX = B
for the given vector B.
"
#
" #
1 −1
2
(a) A =
;B=
2
2
2



5
1 1 2






(b) A =  2 1 1 ; B =  2
−1
−2 2 1




19. Consider a matrix A ∈ M (5, 8). Suppose that the rank of A is 4. What is the dimension
of the null space? Is there a vector B ∈ R4 for which the equation AX = B has no
solution? If no, then prove it. If yes, then give an example of such a matrix A and a
vector B
7
20. The reduced row echelon form of the matrix A that is associated to the homogeneous
system of equations
5x + 4y + 3z + 6w = 0
x
+ 3y + 2z + 2w = 0
3x − 2y −
is

z
1 0

 0 1

0 0
+ 2w = 0
1
11
7
11
10
11
4
11
0
0




(a) Determine the solution set.
(b) Give a spanning set for the column space.
21. Solve the system of equations
x
+ 2y
=
3
4x + 8y + 2z = 14
x
22. Compute the matrix product:
"
+ 2y +
1/5 4/5
0
# "
·
1
z
=
4
5 −4 20
0
1
#
0
23. (a) Write the augmented matrix for the system of equations:
x
+ y = 1
2x − y = 2
(b) Solve the system.
24. Give an example of a linearly dependent set of 3 vectors in M (2, 1) such that any two
vectors taken from the set form a linearly independent set.
25. Give an equation of the form Ax + By + Cz = 0 such that the vectors X = [1, 2, 1]t
and Y = [1, 0, 3]t satisfy the equation.
26. Is the matrix

0 1 2 2 4




A=
0
0
1
2
4


0 0 0 0 1
in reduced row echelon form? If not, then row reduce it. Indicate your steps clearly.
8
27. Determine the image of the unit square (the square with vertices (0, 0)t , (1, 0)t , (1, 1)t ,
(0, 1)t ) under the linear transformation
" # "
#
x
3x − y
L
=
.
y
x − 2y
28. Determine a basis for the null space of the matrix
"
#
3 2 0 2
0 2 3 −2
.
29. The reduced row echelon form of the matrix

−1 −2 8 1 3

A=
2 4 1 −1
 1
2
4 2 1 −3
is

1 2 0
1
3
1
6

A0 = 
 0 0 1
0 0 0 0
− 53
1
6
0






.

(a) Give a basis for the column space of A.
(b) Give a basis for the row space of A.
(c) Give a basis for the null space of A.
(d) Write the vector [1, 1, 1]t as a linear combination of the basis vectors of the column
space that you found above.
(e) The column space defines a plane in M (3, 1). Give an equation for this plane in
the form Ax + By + Cz = 0 where A, B, and C are constants.
30. A matrix A ∈ M (4, 7) has rank 2. What is the dimension of the null space? Give an
example of such a matrix, and a vector B ∈ M (4, 1) such that there is no solution, X,
to the equation AX = B. You may choose A to be in reduced row echelon form.
31. Without performing any computation, explain why {[1, 2, 4], [2, 3, 5], [5, 8, 6], [−10, 4, 7]}
is not linearly independent.
9
32. Determine the row space, column space, and null

2 1 1

A=
 1 3 2
5 0 1
33. Consider the matrix

space of the matrix:




2 4 −3 5

A=
 1 1


1 
.
4 6 −1 7
1
(a) What is the rank of A? (b) Show that there is a vector B ∈ M (3, 1) for which
there is no solution to the equation AX = B.
Let
"
A=
2 −5
3
#
1
(a) Compute A2 .
(b) Use the results above to compute f (A) where
f (x) = x2 − 3x + 17.
34. Compute the eigenvalues and corresponding eigenvectors for the matrix
"
#
13 2
A=
.
4 11
35. Let
"
A=
3 −4
2 −6
#
.
(a) Determine the eigen-values and associated eigen-vectors.
(b) Find matrices P and D such that D is diagonal, P is invertible and
P −1 AP = D.
(c) Compute A5 .
10
36. Consider the quadratic form
q(x, y) = 2x2 − 6xy + 10y 2 .
(a) Give a (2 × 2)-symmetric matrix A such that
"
# " #
a b
x
q(x, y) = [x, y] ·
·
.
b d
y
(b) Determine the eigen-values and associated eigen-vectors of A.
(c) Find a diagonal matrix D and and invertible matrix P so that P −1 AP = D.
(d) describe the quadratic curve q(x, y) = 1.
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