(2) Let V be a subspace of Rn and T : R" → Rm be a linear transformation. Show that T(V) = {T(v) : 7 € V} is a subspace of Rm and dim(V) ≥ dim(T(V)). (Hint: let {₁,...,} be a basis for V.)

4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.

Let T: R¹ → R" be a linear transformation, and let {₁,...,} be a set vectors in Rr. Prove that if {T(v₁),...,T(up)} is linearly independent, then {₁,...,Up} is linearly independent.

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