Question: 1. Consider the linear transformation 7 : 13 -> R given by CI - 12 -213 T 3r2 + 613 3x1 -212 - 413 (a)

1. Consider the linear transformation 7 : 13 -> R given by CI - 12 -213 T 3r2 + 613 3x1 -212 - 413 (a) Find a basis for the subspace that consists of all I'( R" such that T(a) = 0. Is T' injective? (b) Find a basis for the image of 7. Is T surjective? 2. Consider the following general 3 x 3 matrix b C A = e h (a) Let B be the matrix obtained from A by switching row 2 and row 3. Show via an explicit calculation that det A = - det B. (b) Let C be the matrix obtained from A by replacing R3 (row 3) with R3 + t Rl (row 3 plus t times row 1) for some scalar t. Show via an explicit calculation that det A = det C (hint: expand C along its third row)

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