T u true or false 9 11 6 t u 7 11 2 true explanation

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Unformatted text preview: 4. T (u) = 1. FALSE correct −7 10 5. T (u) = True or False? −9 11 6. T (u) = −7 11 2. TRUE Explanation: If A is m × n, then A has n columns, so if A has m pivot columns, then m ≤ n. On the other hand, TA : Rn → Rm is one-to-one if and only if the columns of A are linearly independent. Explanation: But the Fundamental Theorem, T is given by the matrix mapping When m < n, however, the columns of A are not linearly independent. For example, the columns of the matrix A= 1 0 01 11 Thus T : x → [T (e1 ) T (e2 ) T (e3 )] x x 2 −4 1 1 x2 . = −1 3 1 x3 are not linearly independent. FALSE . 009 T (u) = 10.0 points and T (e3 ) = 2 , T (e 2 ) = −1 −4 , 3 1 , determine T (u) when 1 1 3 . u= 2 −4 3 Consequently, If T : R3 → R2 is a linear transformation such that T (e 1 ) = 2 −1 T (u) = Consequently, the statement is 4 010 −8 10 1 1 3. 1 2 . 10.0 points If T : R3 → R2 is the linear transformation such that x1 1 −4 2 + x3 + x2 T x2 = x1 1 1 3 x3 determine T (u) when 1 2. u= 3 1. T (u) = −9 10 2. T (u) = −8 11 1. T (u) = −3 9 3. T (u) = −8 correct 10 2. T (u) = −2 8 khan (sak2454) – HW05 – gilbert – (57245) 3. T (u) = −4 9 4. T (u) = −4 8 5. T (u) = −3 correct 8 6. T (u) = −2 9 Explanation: By definition, and T (e3 ) = −4 , 1 2 , T (e 2 ) = 3 T (e 1 ) = 1 . 1 Thus by the Fundamental Theorem, T is given as a matrix mapping and so T : x → [T (e1) T (e2 ) T (e3 )] x x 2 −4 1 1 x2 , = 311 x3 T (u) = 2 −4 31 Consequently, T (u) = 1 1 2. 1 3 −3 8 . 5...
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This document was uploaded on 02/26/2014.

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