3. Let V and W be n and m dimensional F-vector spaces and let B and A be bases for V and W respectively. Let ³ : V → Fn and 7 : W → Fm denote the coordinate isomorphisms. Let S : V→ W be a linear transformation. (a) Prove that 7 maps Ker S onto null[S] and √ maps imS onto col[S]. ² (b) Conclude that null[S] ≈ (ker S) and col[S] ≈ imS are isomorphisms. (c) Write a statement that connects rank and nullity of [S] to the kernel and image of S.

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