Mathematics 244, section 1 – Linear Algebra Practice True-False Questions for Exam 2 March 19, 2004

For each true statement, give a short proof or example; for each false statement give a reason or a counterexample A) If T : V → W is linear, dim(V) = 5, dim(W) = 7 then T is surjective. B) Every linear T : V → W, where dim(V) = 5, dim(W) = 7, is injective. C) Some linear T : V → W, where dim(V) = 5, dim(W) = 7, is injective. D) If {v_{1}, v_{2}, …, v_{n}} in V is linearly dependent, and T : V → W is linear, then

{T (v_{1}), …, T (v_{n})}

in W is linearly dependent. E) If T : R^{n} → R^{n} is linear, then Ker(T) ∩ Im(T) = {0}. F) If T : V → V is linear, and β = {v_{1}, …, v_{n}}, B′ = {cv_{1}, …, cv_{n}}, c ≠ 0 ∈ R are bases

for V, then the matrices A, A′ of T with respect to β, β and β′, β′ are equal. G) If A is an n × n matrix, and c is a fixed real number, the set W = {v ∈ R^{n} : Av = cv}

is a vector subspace of R^{n}. H) If T = I_{R^{2}} is the identity mapping and β, β′ are two bases for R^{2}, then the matrix of

T with respect to β, β′ is ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ). I) If T : V → W is an isomorphism, and U ⊆ V is a vector subspace, then dim(U) =

dim(T (U)). J) If v_{1}, …, v_{n} are vectors in a vector space V, and w_{1}, …, w_{n} are vectors in a vector

space W, then there always exists some linear T : V → W satisfying T (v_{i}) = w_{i} for all i = 1, …, n.

K) If T : R^{n} → R^{n} is linear, and Ker(T) + Im(T) = R^{n}, then Ker(T) ∩ Im(T) = {0}.