2. Linear Algebra

Orthogonality — Quiz

Test your understanding of orthogonality with 5 practice questions.

Practice Questions

Question 1

Given a set of vectors $ \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\} $ in an inner product space, what is the most stringent condition for this set to be considered an orthonormal basis?

Question 2

Consider the inner product space $ \mathbb{R}^3 $ with the standard Euclidean inner product. If $ \mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} $ and $ \mathbf{u} = \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix} $, what is their inner product $ \langle \mathbf{v}, \mathbf{u} \rangle $?

Question 3

Let $ W $ be a subspace of an inner product space $ V $. Which of the following statements is always true about the orthogonal complement $ W^{\perp} $?

Question 4

Given a vector $ \mathbf{v} = \begin{pmatrix} 5 \\ 2 \end{pmatrix} $ and a subspace $ W $ spanned by the orthonormal basis $ \{\mathbf{u}_1\} = \left\{ \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{pmatrix} \right\} $. What is the orthogonal projection of $ \mathbf{v} $ onto $ W $?

Question 5

In the Gram-Schmidt process, if we have a set of linearly independent vectors $ \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} $, what is the formula for the third orthogonal vector $ \mathbf{u}_3 $?