Math 636 Assignment 7 - Quiz Component 1. If L : R2 → R3 is the linear mapping defined by L(x1 , x2 ) = (x1 +x2 , 2x1 +2x2 , −x1 −x2 ),
then
(a) nullity(L) = 0
(b) nullity(L) = 1
(c) nullity(L) = 2
(d) nullity(L) = 3
2. If T : P3 (R) → M2×2 (R) is the linear mapping defined by


a
b+c
2
3
T (a + bx + cx + dx ) =
a+b a+c
then
(a) rank(T ) = 0
(b) rank(T ) = 1
(c) rank(T ) = 2
(d) rank(T ) = 3
3. Which of the following is not a linear mapping?
(a) tr : Mn×n (R) → R (called the trace of a matrix) defined by tr(A) = n
X (A)ii i=1 (b) M : P2 (R) → P3 (R) defined by M (a + bx + cx2 ) = a + (b + c)x
(c) D : P3 (R) → P3 (R) defined by D(a + bx + cx2 + dx3 ) = b + 2cx + 3dx2
(d) T : R2 → R2 defined by T (x, y) = (x + y, 1)
For questions 4 - 8, determine if the statement is True or False. You should make sure that
you have a proof of each true statement and a counter example for each false statement.
4. Let L : V → W and M : W → U be linear mappings. If Range(L) = W and
Range(M ) = U, then Range(M â—¦ L) = U.
(a) True.
(b) False.
5. There exists linear mappings L : V → W and M : W → U such that Range(M ) 6= U,
but Range(M â—¦ L) = U.
(a) True.
(b) False. 1 2 6. If {~v1 , . . . , ~vk } spans V and L : V → W is linear, then {L(~v1 ), . . . , L(~vk )} spans W.
(a) True.
(b) False.
7. If L : V → W is a linear mapping such that dim V ≥ dim W, then Range(L) = W.
(a) True.
(b) False.
8. If L : V → W and M : W → U are linear mapping, then ker(L) ⊆ ker(M ◦ L).
(a) True.
(b) False.

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