Homework 12
MATH 231
Due Wednesday, December 10, 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 6.1 in the course textbook:

# 38, 39

Exercise 2.

Complete the following exercises from Section 6.2 in the course textbook:

# 41

Exercise 3.

Let $𝐯=\left[\begin{array}{c}3\\ 4\end{array}\right]$, and let $W=\mathrm{span}\{𝐯\}$. Let $T:{ℝ}^2\to {ℝ}^2$ be given by

$$T(𝐮)={\mathrm{proj}}_W(𝐮)$$

(In #41 in Section 6.2, you established that $T$ is a linear transformation.) Find the matrix $A$ satisfying $T(𝐮)=A𝐮$ for every $𝐮\in {ℝ}^2$.

Exercise 4.

Complete the following exercises from Section 6.3 in the course textbook:

# 1, 3, 5, 7, 31, 32

Exercise 5.

Let $W$ be a subspace of ${ℝ}^n$.

1. Show that $W\subset {(W^{⟂})}^{⟂}$.

2. Show that ${(W^{⟂})}^{⟂}\subset W$. (Hint: Let $w\in {(W^{⟂})}^{⟂})$. Use the orthongal projection theorem to write $w=\widehat{w}+z$, where $\widehat{w}\in W$ and $z\in W^{⟂}$. Argue that $z=\mathrm{𝟎}$.)

Together, parts (a) and (b) show that $W={(W^{⟂})}^{⟂}$. (Fun fact: this fails in infinite dimensional spaces!)

Exercise 6.

This exercise gives another proof of the fact that $W={(W^{⟂})}^{⟂})$.

1. Let $X$ and $Y$ be subspaces of ${ℝ}^n$ with $Y\subset X$. Show that if $dimX=dimY$, then $X=Y$.

2. Combine part (a) of this exercise together with part (a) from Exercise 5 and part (c) of Exercise 32 in Section 6.3 to deduce that if $W$ is a subspace of ${ℝ}^n$, then $W={(W^{⟂})}^{⟂}$.