3. Linear Algebra

Orthogonality — Quiz

Test your understanding of orthogonality with 5 practice questions.

Practice Questions

Question 1

Given an inner product space $V$ and a subspace $W$. If $\mathbf{v} \in V$ and $\mathbf{w} \in W$, what is the relationship between $\mathbf{v} - \text{proj}_W \mathbf{v}$ and $\mathbf{w}$?

Question 2

Given an inner product space $V$ and a subspace $W$ with an orthonormal basis $\{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_k\}$. For any vector $\mathbf{v} \in V$, which of the following expressions correctly represents the orthogonal projection of $\mathbf{v}$ onto $W$?

Question 3

Consider the vectors $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ in $\mathbb{R}^2$ with the standard dot product. Apply the Gram-Schmidt process to find the first orthogonal vector $\mathbf{u}_1$.

Question 4

Let $P$ be the orthogonal projection matrix onto a subspace $W$. Which of the following properties must $P$ satisfy?

Question 5

Suppose $\{\mathbf{u}_1, \mathbf{u}_2\}$ is an orthonormal basis for a subspace $W$ of $\mathbb{R}^3$. If $\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$, $\mathbf{u}_1 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix}$, and $\mathbf{u}_2 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$, calculate the orthogonal projection of $\mathbf{v}$ onto $W$.