Linear Algebra and Its Applications, Exercise 3.4.28

Exercise 3.4.28. Given the plane $x_1 + x_2 + x_3 = 0$ and the following vectors

$\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \qquad \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} \qquad \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$

in the plane, find an orthonormal basis for the subspace represented by the plane. Report the dimension of the subspace and the number of nonzero vectors produced by Gram-Schmidt orthogonalization.

Answer: We start with the vector $a_1 = (1, -1, 0)$ and normalize it to create $q_1$ :

$\|a_1\|^2 = 1^2 + (-1)^2 + 0^2 = 1 + 1 = 2$

$q_1 = a_1/\|a_1\| = \frac{1}{\sqrt{2}} a_1 = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{bmatrix}$

We then take the second vector $a_2 = (0, 1, -1)$ and create a second orthogonal vector $a_2'$ by subtracting from $a_2$ its projection on $q_1$ :

$a_2' = a_2 - (q_1^Ta_2)q_1$

$= a_2 - \left[ \frac{1}{\sqrt{2}} \cdot 0 + (-\frac{1}{\sqrt{2}}) \cdot 1 + 0 \cdot (-1) \right]q_1 = a_2 - (-\frac{1}{\sqrt{2}})q_1 = a_2 + \frac{1}{\sqrt{2}}q_1$

$= \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} + \frac{1}{\sqrt{2}} \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} + \begin{bmatrix} \frac{1}{2} \\ -\frac{1}{2} \\ 0 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \\ \frac{1}{2} \\ -1 \end{bmatrix}$

We then normalize $a_2'$ to create $q_2$ :

$\|a_2'\|^2 = (\frac{1}{2})^2 + (\frac{1}{2})^2 + (-1)^2 = \frac{1}{4} + \frac{1}{4} + 1 = \frac{3}{2}$

$q_2 = a_2'/\|a_2'\| = a_2'/\sqrt{\frac{3}{2}} = \frac{\sqrt{2}}{\sqrt{3}} \begin{bmatrix} \frac{1}{2} \\ \frac{1}{2} \\ -1 \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2\sqrt{3}} \\ \frac{\sqrt{2}}{2\sqrt{3}} \\ -\frac{\sqrt{2}}{\sqrt{3}} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} \\ -\frac{2}{\sqrt{6}} \end{bmatrix}$

Finally, we take the third vector $a_3 = (1, 0, -1)$ and attempt to create another orthogonal vector $a_3'$ by subtracting from $a_3$ its projections on $q_1$ and $q_2$ :

$a_3' = a_3 - (q_1^Ta_3)q_1 - (q_2^Ta_3)q_2$

$= a_3 - \left[ \frac{1}{\sqrt{2}} \cdot 1 + (-\frac{1}{\sqrt{2}}) \cdot 0 + 0 \cdot (-1) \right]q_1- \left[ \frac{1}{\sqrt{6}} \cdot 1 + \frac{1}{\sqrt{6}} \cdot 0 + (-\frac{2}{\sqrt{6}}) \cdot (-1) \right] q_2$

$= a_3 - \frac{1}{\sqrt{2}}q_1 - \frac{3}{\sqrt{6}}q_2 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} - \frac{1}{\sqrt{2}} \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{bmatrix} - \frac{3}{\sqrt{6}} \begin{bmatrix} \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} \\ -\frac{2}{\sqrt{6}} \end{bmatrix}$

$= \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} - \begin{bmatrix} \frac{1}{2} \\ -\frac{1}{2} \\ 0 \end{bmatrix} - \begin{bmatrix} \frac{3}{6} \\ \frac{3}{6} \\ -\frac{6}{6} \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} - \begin{bmatrix} \frac{1}{2} \\ -\frac{1}{2} \\ 0 \end{bmatrix} - \begin{bmatrix} \frac{1}{2} \\ \frac{1}{2} \\ -1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

Since $a_3' = 0$ we cannot create a third orthogonal vector to $q_1$ and $q_2$ . The vectors

$q_1 = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{bmatrix} \qquad q_2 = \begin{bmatrix} \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} \\ -\frac{2}{\sqrt{6}} \end{bmatrix}$

are an orthonormal basis for the subspace, and the dimension of the subspace is 2.

(In hindsight we could have predicted this result by inspecting the original vectors $a_1$ , $a_2$ , and $a_3$ and noticing that $a_3 = a_1 + a_2$ . Thus only $a_1$ and $a_2$ were linearly independent, $a_3$ being linearly dependent on the first two vectors, so that only two orthonormal basis vectors could be created from the three vectors given.)

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books.

Linear Algebra and Its Applications, Exercise 3.4.28

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