1.9 Exercises

20 points · 4 extra credit

The Matrix of a Linear Transformation

1. 1. (2 pts) Find the standard matrix of the linear transformation $T : \mathbb{R}^2 \to \mathbb{R}^3$ given by $T(x_1, x_2) = (x_1 + 2x_2,\; 3x_1 - x_2,\; x_2)$.

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2. 2. (3 pts) Find the standard matrix of the linear transformation that reflects each point of $\mathbb{R}^2$ across the line $y = x$.

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3. 3. (3 pts) Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \end{bmatrix}$. Determine whether $T$ is onto $\mathbb{R}^2$ and whether $T$ is one-to-one. Justify using pivots.

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4. 4. (3 pts) Let $T(\mathbf{x}) = A\mathbf{x}$ with $A = \begin{bmatrix} 1 & 2 \\ 2 & 5 \\ 1 & 1 \end{bmatrix}$. Determine whether $T$ is one-to-one and whether $T$ is onto $\mathbb{R}^3$. Justify using pivots.

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5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) A linear transformation $T$ is one-to-one if and only if the columns of its standard matrix are linearly independent.

   (b) A linear transformation $T : \mathbb{R}^5 \to \mathbb{R}^3$ can be one-to-one.

   (c) A linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^5$ can be onto.

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6. 6. (3 pts) Suppose $T : \mathbb{R}^4 \to \mathbb{R}^3$ is linear and onto. Explain what this forces about the number of pivots in its standard matrix, and use that to decide whether $T$ can be one-to-one.

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7. 7. (3 pts) Let $T(\mathbf{x}) = A\mathbf{x}$. Prove that $T$ is one-to-one if and only if the equation $A\mathbf{x} = \mathbf{0}$ has only the trivial solution.

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8. 8. (4 pts ••) Prove: if $T : \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation whose standard matrix is $A$, then

   (a) $T$ onto implies $n \ge m$;

   (b) $T$ one-to-one implies $n \le m$.

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