Question

(16) Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ and $R: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the following transformations: $T(\vec{v}) = \begin{bmatrix} 3x - 2y \ z + y \end{bmatrix}$, $R(\vec{v}) = \begin{bmatrix} y + 3 \ x + z \ z - 2y \end{bmatrix}$, for every vertex $\vec{v} = [x, y, z]^T \in \mathbb{R}^3$. Is T a linear transformation? Is R a linear transformation? Fully justify your answer.

          (16) Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ and $R: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the following transformations:
$T(\vec{v}) = \begin{bmatrix} 3x - 2y \ z + y \end{bmatrix}$,
$R(\vec{v}) = \begin{bmatrix} y + 3 \ x + z \ z - 2y \end{bmatrix}$,
for every vertex $\vec{v} = [x, y, z]^T \in \mathbb{R}^3$. Is T a linear transformation? Is R a linear transformation? Fully justify your answer.

(16) Let T: ℝ^3 →ℝ^2 and R: ℝ^3 →ℝ^3 be the following transformations:
T(v⃗) =
< b m a t r i x >,
R(v⃗) =
< b m a t r i x >,
for every vertex v⃗ = [x, y, z]^T ∈ℝ^3. Is T a linear transformation? Is R a linear transformation? Fully justify your answer.

Added by Amy D.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

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Solved by Expert Suzanne W.

Step 1

A transformation L is linear if it satisfies the following two properties for all vectors u, v and scalar c: Show more…

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Let T: R^3 -> R^2 and R: R^3 -> R^3 be the following transformations: T([x,y,z]) = [3x - y, z] and R([x,y,z]) = [x + 2z + y, 2y]. Is T a linear transformation? Is R a linear transformation? Fully justify your answer.