MATH 304
1.
Midterm 2 Review
Find a basis for the null space of A, where

1
A = 2
3
2
3
4
3
4
5

4
5
6
Page 1
MATH 304
2.
Midterm 2 Review
a) (5 pts.) Given a vector space V and two subspaces U, W of V , is the set
U ∪ V = {v ∈ V : v ∈ U or v ∈ W }
a subspace of V ? Why or why not?
b) (5pts.) Let
x
Q={
∈ R2 : x ≥ 0 and y ≥ 0}.
y
Is Q a subspace of R2 ? Explain.
c) (5 pts.) Let
x
T ={
∈ R2 : xy ≥ 0}.
y
Is T a subspace of R2 ? Explain.
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MATH 304
Midterm 2 Review
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March 23, 2016
3.
(5 pts.) Let V and W be vector spaces and L : V → W a linear transformation. Let
BV = {e1 , . . . , en }
and BW = {f1 , . . . , fn }
be a choice of bases for V and W respectively. Let A be the matrix representation of L with respect
to the bases BV and W. Which of the following scenarios are possible? Explain why or why not.
Give examples if you think it is possible.
a) (3 pts.) n = 3, m = 4, and {v1 , v2 , v3 } is a linearly independent set of vectors.
b) (3 pts.) n = 5, m = 3, rank(A) = 4.
c) (3 pts.) n = 5, m = 3 and for some u1 , u2 , u3 , u4 ∈ V , one has that Span(L(u1 ), L(u2 ), L(u3 ), L(u4 )) =
W.
d) (3 pts.) n = 5, m = 3, and rank(A) = 2, nullity(A) = 1.
e) (3 pts.) n = 100, m = 200, Ker(L) = V and Im(L) = W .
MATH 304
Midterm 2 Review
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March 23, 2016
4.
Let

1
−2

A=
0
3
2 0 2
−5 1 −1
−3 3 4
6 0 −7

5
−8
.
1
2
a) (7 pts.) Find a basis for the null space N (A).
b) (3 pts.) What is the dimension of the column space of A?
c) (2 pts.) What is the nullity of A?
d) (3 pts.) Find a basis for the row space of A.
MATH 304
Midterm 2 Review
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March 23, 2016
5.
(15 pts.) Let P3 be the set of polynomials of degree less than or equal to 3, i.e.
P = {p(x) = a0 + a1 x + a2 x2 + a3 x3 : a0 , a1 , a2 , a3 ∈ R}.
The set B1 = {1, x, x2 , x3 } is a basis for P3 .
a) Show that B2 = {1, (1 + x), (1 + x)2 , (1 + x)3 } is a basis for P3 .
b) Write the coordinates of elements of B2 in terms of the basis vectors B1 .
MATH 304
Midterm 2 Review
Page 6
March 23, 2016
6.
(15 pts.) Let P3 be the set of polynomials of degree less than or equal to 3. Assume that B2 =
{1, (1 + x), (1 + x)2 , (1 + x)3 } is a basis of P3 . Express
p(x) = 1 + 3x + 4x2 − x3
with respect to the basis B2 , i.e. what are the coordinates of p in B2 .
MATH 304
Midterm 2 Review
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March 23, 2016
7.
(15 pts.) Let f (x) = x3 , g(x) = ex and h(x) = e−x , be vectors in V = C ∞ (R), the space of
continuous functions on the real line. Show that these vectors linearly independent.
MATH 304
Midterm 2 Review
Page 8
March 23, 2016
8.
(15 pts.) Which of the following are linear functions, explain.
x
x
a) T : R2 → R2 where T
is the reflection of the point
with respect to the line x + y = 1.
y
y
b)
T : R3 → R3
v 7→ v + v0
 
1
where v0 = 2.
3
c) T : R → R, where T (t) is the number of people alive on earth at time t.
d) T : V → R where T (v) = 0 for every v ∈ V with V a vector space.
e) T : R → R defined with T (x) = |x|.
MATH 304
Midterm 2 Review
Page 9
March 23, 2016
9.
Let

1
−2

A=
0
3
2 0 2
−5 1 −1
−3 3 4
6 0 −7

5
−8
.
1
2
One has that after row operations the following reduced row echelon form,


1 0 2 0 1
row operations 0 1 −1 0 1


A
−→
0 0 0 1 1 .
0 0 0 0 0
Find a basis of Col(A). Explain why it is a basis.
MATH 304
Midterm 2 Review
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March 23, 2016
10.
Show that if A is a nonsingular matrix then det(A−1 ) =
1
det(A) .
MATH 304
Midterm 2 Review
Page 11
March 23, 2016
11.
Let L : R2 → R2 be a linear function such that with respect to the standard basis it is given by the
matrix
3 4
A=
.
0 1
What is the matrix of the same linear representation with respect to the basis given by {v1 , v2 }
where
3
−1
v1 =
and
v2 =
.
−1
3
MATH 304
12.
Midterm 2 Review
Page 12
Let P3 be the vector space of polynomials of degree less than or equal to 3, and let
D : P3 → P3
p(x) 7→ p0 (x)
be the differentiation map, e.g. if p(x) = 2 + 3x − 4x2 + x3 , then D(p) = q with q(x) = 3 − 8x + 3x2 .
Write down the matrix of this linear transformation with respect to the basis {1, x, x2 , x3 }.
MATH 304
Midterm 2 Review
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March 23, 2016
13.
What is the matrix representation of the linear operator on R2 that rotates every vector by π/2 in
the counterclockwise direction, with respect to the standard basis.
MATH 304
Midterm 2 Review
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March 23, 2016
14.
v
Let v = 0 and let w = Rotπ/2 v, i.e. w is the vector you obtain if you rotate v by π/2 radians in
v1
the counterclockwise direction. {v, w} is a basis of R2 .
Let T be the linear map from R2 to R2 that reflects the plane with respect to the line going through
the origin that contains v.
a) What is the matrix of the linear transformation T with respect to the basis {v, w}?
b) Express the same linear transformation T by a matrix in the standard basis.