(1) Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be defined by \( T(x, y, z) = (2x - z, y + z, y - 3x) \). Let \( \alpha = \{ (1, 1, 0), (1, 0, 1), (0, 1, 1) \} \) and \( \beta = \{ (-1, 1, 1), (1, -1, 1), (1, 1, -1) \} \).(a) Find the change of basis matrix from \(\alpha\) to \(\beta\).(b) Prove that \( T \) is linear.(c) Find the Kernel and Image of \( T \).(d) Compute \([T]_{\alpha}^{\alpha}, [T]_{\beta}^{\alpha}, [T]_{\beta}^{\beta}\) and find an invertible matrix \( Q \) such that \([T]_{\alpha}^{\alpha} = Q^{-1}[T]_{\beta}^{\beta}Q\). Can you identify \( Q \)?(e) Let \( v = (2, -1, 5) \). Compute \( T(v) \) three ways: directly from the definition and using \([T]_{\alpha}^{\alpha}\) and \([T]_{\beta}^{\beta}\). Do they agree? Why or why not?

As per our company guidelines we are supposed to do Fisrt 3 subparts . Kindly repost the others in…

Let T : R3 →→ R³ be defined by T(x, y, z) {(1,1,0), (1,0, 1), (0, 1, 1)} and ß = {(-1,1,1), (1, –1, 1), (1, 1, –1)}. (a) Find the change of basis matrix from a to ß. (b) Prove that T is linear. (c) Find the Kernel and Image of T. (2x – z, y + z, y – 3x). Let a =

Let T : R3 →→ R³ be defined by T(x, y, z) {(1,1,0), (1,0, 1), (0, 1, 1)} and ß = {(-1,1,1), (1, –1, 1), (1, 1, –1)}. (a) Find the change of basis matrix from a to ß. (b) Prove that T is linear. (c) Find the Kernel and Image of T. (2x – z, y + z, y – 3x). Let a =

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

(1) Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be defined by \( T(x, y, z) = (2x - z, y + z, y - 3x) \). Let \( \alpha = \{ (1, 1, 0), (1, 0, 1), (0, 1, 1) \} \) and \( \beta = \{ (-1, 1, 1), (1, -1, 1), (1, 1, -1) \} \).

(a) Find the change of basis matrix from \(\alpha\) to \(\beta\).

(b) Prove that \( T \) is linear.

(c) Find the Kernel and Image of \( T \).

(d) Compute \([T]_{\alpha}^{\alpha}, [T]_{\beta}^{\alpha}, [T]_{\beta}^{\beta}\) and find an invertible matrix \( Q \) such that \([T]_{\alpha}^{\alpha} = Q^{-1}[T]_{\beta}^{\beta}Q\). Can you identify \( Q \)?

(e) Let \( v = (2, -1, 5) \). Compute \( T(v) \) three ways: directly from the definition and using \([T]_{\alpha}^{\alpha}\) and \([T]_{\beta}^{\beta}\). Do they agree? Why or why not?

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Let T: R2 R³ be given by T(x, y) = (x+y, x, y). Let B = {(1,−1), (0, 1)} and B' = {(1, 1, 0), (0, 1, 1), (1, 0, 1)}. Let v = (5, 4). a) Find the standard matrix for T. b) Find the matrix for T relative to B and B'. c) Find T(v) using the matrix relative to B and B' (the matrix you found in PART b).
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