solq3 - Solutions to quiz # 3 (September 23) 1. Bringing...

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Unformatted text preview: Solutions to quiz # 3 (September 23) 1. Bringing the matrix to the 1 a 3 reduced row-echelon form, we obtain 11 1 1 1 2 4 −→ 0 2 − a 4 − a . 3a 0 0 a−3 If a = 2, 3 we can proceed further 1 1 1 1 0 2 − a 4 − a −→ 0 0 0 a−3 0 1 1 0 1 and the rank of the matrix is 3. If a = 2, we have the matrix 11 1 11 0 0 2 −→ 0 0 0 0 −1 00 and the rank of the matrix is 2. If a = 3, we have the matrix 11 0 −1 00 0 100 0 −→ 0 1 0 1 001 11 4−a −→ 0 1 2−a 1 00 1 10 −→ 0 1 1 0 00 0 1 0 2 −1 0 and the rank of the matrix is 2. Answer: The rank is 2 when a = 2 or when a = 3. 2. We observe that vector vector 1 0 is mapped to vector 0 −1 −1 . Hence the matrix of the transformation is 0 y 0 1 y=x −1 0 and vector 0 −1 1 0 x 0 0 −1 Answer: The matrix of the transformation is 1 0 −1 −1 . 0 −1 . 0 0 1 is mapped to ...
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