## Linear Algebra and Its Applications, Exercise 3.4.13

Exercise 3.4.13. Given the vectors $a = \begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix} \quad b = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \quad c = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$

and the matrix $A$ whose columns are $a$, $b$, and $c$, use Gram-Schmidt orthogonalization to factor $A = QR$.

Answer: We first choose $a' = a$. We then have $b' = b - \frac{(a')^Tb}{(a')^Ta'}a' = b - \frac{0 \cdot 0 + 0 \cdot 1 + 1 \cdot 1}{0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1}a' = b - \frac{1}{1}a' = b - a'$ $= \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} - \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$

We then have $c' = c - - \frac{(a')^Tc}{(a')^Ta'}a' - \frac{(b')^Tc}{(b')^Tb'}b' = c - \frac{0 \cdot 1 + 0 \cdot 1 + 1 \cdot 1}{0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1}a' - \frac{0 \cdot 1 + 1 \cdot 1 + 0 \cdot 1}{0 \cdot 0 + 1 \cdot 1 + 0 \cdot 0}b'$ $= c - \frac{1}{1}a' - \frac{1}{1}b' = c - a' - b'$ $= \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} - \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} - \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$

We have $\|a'\| = \|b'\| = \|c'\| = 1$, so $q_1 = a'$, $q_2 = b'$, and $q_3 = c'$. The matrix $Q$ is then $Q = \begin{bmatrix} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{bmatrix}$

The matrix $R$ is then $R = \begin{bmatrix} q_1^Ta&q_1^Tb&q_1^Tc \\ 0&q_2^Tb&q_2^Tc \\ 0&0&q_3^Tc \end{bmatrix}$ $= \begin{bmatrix} (0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1)&(0 \cdot 0 + 0 \cdot 1 + 1 \cdot 1)&(0 \cdot 1 + 0 \cdot 1 + 1 \cdot 1) \\ 0&(0 \cdot 0 + 1 \cdot 1 + 0 \cdot 1)&(0 \cdot 1 + 1 \cdot 1 + 0 \cdot 1) \\ 0&0&(0 \cdot 1 + 0 \cdot 1 + 1 \cdot 1) \end{bmatrix}$ $= \begin{bmatrix} 1&1&1 \\ 0&1&1 \\ 0&0&1 \end{bmatrix}$

The product of the two matrices is then $QR = \begin{bmatrix} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{bmatrix} \begin{bmatrix} 1&1&1 \\ 0&1&1 \\ 0&0&1 \end{bmatrix} = \begin{bmatrix} 0&0&1 \\ 0&1&1 \\ 1&1&1 \end{bmatrix} = A$

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 .

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