HW05-solutions (2)

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 02/26/2014.

Ask a homework question - tutors are online