MATH235: Linear Algebra 2
Practice Problems for Chapter 2: Linear Transformations
Instructions
• These problems are meant to give you an opportunity to practice the material covered in class
and in the course notes. They are ungraded: you do not have to submit them. However, we
highly recommend that you attempt all of them and write up full solutions to each.
• You are welcome to discuss these problems with your instructor, classmates, and on Piazza.
Computational Problems
Q1. Let V and W be vector spaces over F and let
L(V, W ) = {L : V → W : L is a linear map}
be the set of all linear maps from V to W . For L, T ∈ L(V, W ) and α ∈ F, define the functions
L + T : V → W and αL : V → W by
(L + T )( #»
v ) = L( #»
v ) + T ( #»
v)
and
(αL)( #»
v ) = α(L( #»
v )).
Show that these operations make L(V, W ) into a vector space over F. [Note: As a first step,
you should prove that L + T and αL are in fact linear maps!]
Q2. Determine if L : V → W is a linear map.
(a) L : C → C, L(z) = Re(z) (the real part of z).
ab
a−b
a
.
=
(b) L : F2 → M2×2 (F), L
b−a
ab
b
(c) L : V → R, L(f ) = f ′ (0), where V = {f : R → R : f is differentiable on R}.
p(1)
p′ (1)
.
(d) L : P2 (R) → M2×2 (R), L(p(x)) = ′′
p (1) p′ (1) − p′ (0)
(e) L : Pn (F) → Pn (F), L(p(x)) = p(x + 1).
(f) L : Mn×n (F) → Pn (F), L(A) = CA (x) (the characteristic polynomial of A).
(g) L : Mn×n (F) → F, L(A) = det(A).
(h) L : Mn×n (F) → Mn×n (F), L(A) = AB − BA, where B is a fixed matrix in Mn×n (F).
(i) L : Mm×n (F) → Mn×m (F), L(A) = AT .
Q3. For each linear map L : V → W given below, (i) find a basis for Range(L); (ii) determine
rank(L), and (iii) determine if L is surjective.
(a) L : P3 (R) → P2 (R), L(p(x)) = p′ (x).


a+b
a
b
(b) L : M2×2 (F) → F3 , L
=  b + c .
c d
c+d
a + 2b
b
2
(c) L : P2 (F) → M2×2 (F), L(a + bx + cx ) =
.
b + c 2a + b
1
MATH235: Linear Algebra 2
Practice Problems for Chapter 2: Linear Transformations
 
a
3


b  = (a − c) + (b + c)x + (a + b)x2 .
(d) L : F → P2 (F), L
c
R1
(e) L : P3 (R) → R, L(p(x)) = 0 p(x) dx.
Q4. For each linear map L : V → W given below, (i) find a basis for Ker(L); (ii) determine
nullity(L), and (iii) determine if L is injective.
(a) L : P3 (R) → P2 (R), L(p(x)) = p′ (x).
(b) L : M2×2 (F) → M2×2 (F), L(A) = A + AT .
a + 2b
b
2
(c) L : P2 (F) → M2×2 (F), L(a + bx + cx ) =
.
b + c 2a + b
 
a
(d) L : F3 → P2 (F), L  b  = (a − c) + (b + c)x + (a + b)x2 .
c
a
= a cos(x) + b sin(x).
(e) L : R2 → C([0, 1]), L
b
Q5. For each linear map L : V → W given below, determine if L is an isomorphism, and if it is,
determine L−1 .
a+b
2a − b + c
2
.
(a) L : P2 (F) → M2×2 (F), L(a + bx + cx ) =
a + 2b + c
3a − b
(b) L : Pn (F) → Fn+1 , L(p(x)) = [p(x)]S , where S is the standard basis for Pn (F).


p(0)
(c) L : P2 (R) → R3 , L(p(x)) =  p′ (0) .
p′′ (0)
(d) L : M1×2 (F) → P1 (F), L a b = (a + b) + (a − b)x.
(e) L : Mm×n (F) → Mn×m (F), L(A) = AT .
Q6. Find C [L]B the matrix of the linear map L : V → W with respect to the ordered bases B and
C of V and W , respectively.
(a) L : P3 (F) → P2 (F), L(p(x)) = p′ (x); B = {1, x, x2 , x3 }, C = {1, x, x2 }.
a b
a+b
2
(b) L : M2×2 (F) → F , L
=
; B = standard basis for M2×2 (F), C =
c d
c+d
1
1
,
.
1
−1
a b
(c) L : P1 (F) → M2×2 (F), L(a + bx) =
; B = {1 + x, 1 − x},
−b a
1 −1
1 0
1 0
0 0
C=
,
,
,
.
0 0
0 1
0 −1
−1 1
2
MATH235: Linear Algebra 2
Practice Problems for Chapter 2: Linear Transformations
1 0
0 1
0 0
0 0
2 1
1 0
0 2
0 2
Q7. Let S =
,
,
,
and B =
,
,
,
0 0
0 0
1 0
0 1
0 0
3 1
−3 2
3 1
be bases for M2×2 (F). Find the change of coordinate matrices S IB and B IS
Q8. Determine which of the following real vector spaces are isomorphic to which. Give an isomorphism between each isomorphic pair.
(a) R.
(b) R2 .
(c) R3 .
(d) P2 (R).
(e) P3 (R).
(f) M1×3 (R).
(g) M3×1 (R).
(h) M2×2 (R).
(i) L = L(R2 , R2 ).
(j) D = {A ∈ M2×2 (R) : A is a diagonal matrix}.
(k) ∆ = {A ∈ M2×2 (R) : A is an upper-triangular matrix}.
1 1
1 1
A .
(l) M = A ∈ M2×2 (R) : A
=
1 1
1 1
(m) S = {A ∈ M3×3 (R) : A = AT }.
(n) W = {A ∈ M3×3 (R) : A = −AT }.
(o) T = {A ∈ M3×3 (R) : tr(A) = 0}.
(p) U = {p(x) ∈ P2 (R) : p′ (0) = p(1) = 0}.
(q) F = C([0, 1]) = {f : [0, 1] → R : f is continuous on [0, 1]}.
Conceptual Problems
Q1. Let L : V → W and T : V → W be linear maps. Suppose that B = { #»
v 1 , . . . , #»
v n } is a basis
#»
#»
for V . Show that L = T if and only if L( v i ) = T ( v i ) for all i. [Note: Equality of linear
maps is equality of functions. That is, L = T means L and T have the same domain V , and
L( #»
v ) = T ( #»
v ) for all #»
v ∈ V .]
Q2. Let L : V → W be a linear map. Suppose that B = { #»
v 1 , . . . , #»
v n } is a basis for V . Show that
#»
#»
Range(L) = Span{L( v 1 ), . . . , L( v n )}.
Q3. Let L : V → W be a linear map. Suppose that { #»
v 1 , . . . , #»
v k } is a linearly independent set of
vectors in V .
(a) Show that if L is injective, then {L( #»
v 1 ), . . . , L( #»
v k )} is linearly independent.
(b) Is part (a) true if L is not injective? Prove/disprove.
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MATH235: Linear Algebra 2
Practice Problems for Chapter 2: Linear Transformations
Q4. Let L : V → W be a linear map between finite-dimensional vector spaces.
(a) Show that if dim V < dim W then L cannot be surjective.
(b) Show that if dim V > dim W then L cannot be injective.
Q5. (a) Give an example of a linear map L : V → W that is injective but not surjective.
(b) Give an example of a linear map L : V → W that is surjective but not injective.
(c) Let L : V → W be a linear map between finite-dimensional vector spaces. Suppose that
dim(V ) = dim(W ). Show that L is injective if and only if L is surjective.
Q6. Let L : V → W be a linear map between finite-dimensional vector spaces.
(a) Prove that L is injective if and only if nullity(L) = 0.
(b) Prove that L is surjective if and only if rank(L) = dim W .
Q7. Let V be an n-dimensional vector space over F, and let L : V → F be a linear map. Determine
nullity(L) and rank(L) in terms of n. [Hint: You will want to consider two cases.]
Q8. Let L : V → W be a linear map between finite-dimensional vector spaces. Suppose that B
and C are ordered bases for V and W , respectively. Prove that L is an isomorphism if and
only if the matrix C [L]B is invertible. In this case, show that B [L−1 ]C = (C [L]B )−1 .
Q9. Let V and W be finite-dimensional vector spaces over F, and let dim V = n and dim W = m.
Prove that L(V, W ) is isomorphic to Mm×n (F).
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