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(4) V2 = A basis for R³ consists of the three vector V₁ = , V3 = 1 2 3 " (You do not have to verify that is a basis for R³.) There is a linear transformation T : R³ → R³ which satisfies T(v₁) = 2v₁, T(v₂) = −5v3, T(V3) = V1 — 2Vv2. Compute the determinant of T.

(4) V2 = A basis for R³ consists of the three vector V₁ = , V3 = 1 2 3 " (You do not have to verify that is a basis for R³.) There is a linear transformation T : R³ → R³ which satisfies T(v₁) = 2v₁, T(v₂) = −5v3, T(V3) = V1 — 2Vv2. Compute the determinant of T.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 4:** A basis \( \beta \) for \( \mathbb{R}^3 \) consists of the three vectors \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \), \( \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \), \( \mathbf{v}_3 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \). (You do not have to verify that \( \beta \) is a basis for \( \mathbb{R}^3 \).) There is a linear transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) which satisfies \( T(\mathbf{v}_1) = 2\mathbf{v}_1 \), \( T(\mathbf{v}_2) = -5\mathbf{v}_3 \), \( T(\mathbf{v}_3) = \mathbf{v}_1 - 2\mathbf{v}_2 \). Compute the determinant of \( T \).
Transcribed Image Text:**Problem 4:** A basis \( \beta \) for \( \mathbb{R}^3 \) consists of the three vectors \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \), \( \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \), \( \mathbf{v}_3 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \). (You do not have to verify that \( \beta \) is a basis for \( \mathbb{R}^3 \).) There is a linear transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) which satisfies \( T(\mathbf{v}_1) = 2\mathbf{v}_1 \), \( T(\mathbf{v}_2) = -5\mathbf{v}_3 \), \( T(\mathbf{v}_3) = \mathbf{v}_1 - 2\mathbf{v}_2 \). Compute the determinant of \( T \).
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