Homework 4
MATH 231
Due Wednesday, October 15, 2025
 

Instructions. We wiill have a quiz in class on the due date based on the content from the assignment. See the back of the textbook for solutions and hints for odd-numbered problems. Click here for a pdf version of the homework.

Exercise 1.

Complete the following exercises from Section 1.8 in the course textbook:

# 1, 3, 5, 9, 11, 17, 19, 32, 38, 41

Exercise 2.

Complete the following exercises from Section 1.9 in the course textbook:

# 1, 3, 4, 5, 6, 15, 17, 33, 37, 38, 39, 40, 44

Exercise 3.

Complete the following exercises from Section 2.1 in the course textbook:

# 7–10, 12, 13, 25, 28, 29, 33, 35

Exercise 4.

Suppose $T:{ℝ}^2\to {ℝ}^3$ is a linear transformation such that $T({𝐞}_1)=\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right]$ and $T({𝐞}_2)=\left[\begin{array}{c}-1\\ -3\\ 2\end{array}\right]$.

1. Find a formula for $T(𝐱)$, that is, compute $T\left(\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\right)$.

2. Find the standard matrix for $T$.

Exercise 5.

Suppose $T:{ℝ}^3\to {ℝ}^2$ is a linear transformation such that

$$T\left(\left[\begin{array}{c}1\\ -3\\ 7\end{array}\right]\right)=T\left(\left[\begin{array}{c}5\\ 2\\ 0\end{array}\right]\right)$$

Find a nontrivial solution to $T(𝐱)=\mathrm{𝟎}$.

Exercise 6.

Suppose $T:{ℝ}^n\to {ℝ}^m$ is a linear transformatin. Show that if $m>n$, then $T$ cannot be onto.