1.8 Exercises

20 points · 4 extra credit

Introduction to Linear Transformations

1. 1. (2 pts) Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 1 & -2 \\ 3 & 0 \\ -1 & 4 \end{bmatrix}$. Compute $T\!\left(\begin{bmatrix} 2 \\ 1 \end{bmatrix}\right)$.

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2. 2. (3 pts) Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 1 & 3 \\ 2 & 5 \end{bmatrix}$. Find all $\mathbf{x}$ whose image is $\mathbf{b} = \begin{bmatrix} 5 \\ 8 \end{bmatrix}$.

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3. 3. (3 pts) Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \\ 1 & 3 \end{bmatrix}$. Determine whether $\mathbf{b} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ is in the range of $T$.

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4. 4. (3 pts) Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. Describe the geometric effect of $T$ on $\mathbb{R}^2$, and compute the image of the point $(2, -1)$.

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5. 5. (3 pts) Suppose $T : \mathbb{R}^2 \to \mathbb{R}^2$ is linear with $T(\mathbf{u}) = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$ and $T(\mathbf{v}) = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$. Find $T(3\mathbf{u} - 2\mathbf{v})$ and $T(\mathbf{0})$.

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) Every linear transformation $T$ satisfies $T(\mathbf{0}) = \mathbf{0}$.

   (b) The range of a transformation $T : \mathbb{R}^n \to \mathbb{R}^m$ is a subset of its domain.

   (c) If $T(\mathbf{x}) = A\mathbf{x}$ with $A$ of size $m \times n$, then the domain of $T$ is $\mathbb{R}^m$.

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7. 7. (3 pts) Let $A$ be an $m \times n$ matrix and define $T(\mathbf{x}) = A\mathbf{x}$. Prove that $T$ is a linear transformation, i.e. that $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{u}) = cT(\mathbf{u})$ for all $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ and all scalars $c$.

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8. 8. (4 pts ••) Fix a nonzero vector $\mathbf{b} \in \mathbb{R}^n$ and define the translation $T(\mathbf{x}) = \mathbf{x} + \mathbf{b}$. Prove that $T$ is not a linear transformation. State the general property of linear maps that translations by $\mathbf{b} \neq \mathbf{0}$ violate.

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