Solved Problems

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Solved Problems for Chapter 9

Find an orthonormal basis for the span of the following set of vectors.

$\left [ \begin {array}{r} 3 \\ -4 \\ 0 \end {array} \right ] ,\left [ \begin {array}{r} 7 \\ -1 \\ 0 \end {array} \right ] ,\left [ \begin {array}{r} 1 \\ 7 \\ 1 \end {array} \right ]$

Click the arrow to see answer.

We will use Gram-Schmidt orthogonalization algorithm. $\vec {f}_1=\begin {bmatrix}3\\-4\\0\end {bmatrix}$ $\vec {f}_2=\begin {bmatrix}7\\-1\\0\end {bmatrix}-\frac {25}{25}\begin {bmatrix}3\\-4\\0\end {bmatrix}=\begin {bmatrix}4\\3\\0\end {bmatrix}$ $\vec {f}_3=\begin {bmatrix}1\\7\\1\end {bmatrix}-\frac {-25}{25}\begin {bmatrix}3\\-4\\0\end {bmatrix}-\frac {25}{25}\begin {bmatrix}4\\3\\0\end {bmatrix}=\begin {bmatrix}0\\0\\1\end {bmatrix}$

Normalizing each vector we get: $\left [ \begin {array}{c} \frac {3}{5} \\ -\frac {4}{5} \\ 0 \end {array} \right ] ,\left [ \begin {array}{c} \frac {4}{5} \\ \frac {3}{5} \\ 0 \end {array} \right ] ,\left [ \begin {array}{c} 0 \\ 0 \\ 1 \end {array} \right ]$

Using the Gram Schmidt process find an orthonormal basis for the following span: $\mbox {span} \left \{ \left [ \begin {array}{r} 1 \\ 2 \\ 1 \end {array} \right ] ,\left [ \begin {array}{r} 2 \\ -1 \\ 3 \end {array} \right ] , \left [ \begin {array}{r} 1 \\ 0 \\ 0 \end {array} \right ] \right \}$

Click the arrow to see the answer.

$\left [ \begin {array}{c} \frac {1}{6}\sqrt {6} \\ \frac {1}{3}\sqrt {6} \\ \frac {1}{6}\sqrt {6} \end {array} \right ] ,\left [ \begin {array}{c} \frac {3}{10}\sqrt {2} \\ -\frac {2}{5}\sqrt {2} \\ \frac {1}{2}\sqrt {2} \end {array} \right ] ,\left [ \begin {array}{c} \frac {7}{15}\sqrt {3} \\ -\frac {1}{15}\sqrt {3} \\ -\frac {1}{3}\sqrt {3} \end {array} \right ]$

Write $\vec {x}$ as the sum of a vector in $W$ and a vector in $W^\perp$ if $\vec {x} = \begin {bmatrix}1\\ 5\\ 7\end {bmatrix},\quad W = \mbox {span}\left (\begin {bmatrix}1\\ -2\\ 3\end {bmatrix}, \begin {bmatrix}-1\\ 1\\ 1\end {bmatrix}\right )$

Click the arrow to see the answer.

Observe that the two vectors that span $W$ are orthogonal. We will refer to these vectors as $\vec {f}_1$ and $\vec {f}_2$ . $\vec {w}=\text {proj}_W(\vec {x})=\text {proj}_{\vec {f}_1}(\vec {x})+\text {proj}_{\vec {f}_2}(\vec {x})=\frac {12}{14}\begin {bmatrix}1\\-2\\3\end {bmatrix}+\frac {11}{3}\begin {bmatrix}-1\\1\\1\end {bmatrix}=\frac {1}{21}\begin {bmatrix}-59\\41\\131\end {bmatrix}$ $\vec {w}^{\perp }=\vec {x}-\vec {w}=\frac {1}{21}\begin {bmatrix}80\\64\\16\end {bmatrix}$ $\vec {x}=\vec {w}+\vec {w}^{\perp }$

Write $\vec {x}$ as the sum of a vector in $W$ and a vector in $W^\perp$ if $\vec {x} = \begin {bmatrix}3\\ 1\\ 5\\ 9\end {bmatrix}, \quad W = \mbox {span}\left (\begin {bmatrix}1\\ 0\\ 1\\ 1\end {bmatrix}, \begin {bmatrix}0\\ 1\\ -1\\ 1\end {bmatrix}, \begin {bmatrix}-2\\ 0\\ 1\\ 1\end {bmatrix}\right )$

Click the arrow to see the answer.

Observe that the three vectors that span $W$ form an orthogonal set. We will refer to these vectors as $\vec {f}_1$ , $\vec {f}_2$ and $\vec {f}_3$ . $\vec {w}=\text {proj}_W(\vec {x})=\text {proj}_{\vec {f}_1}(\vec {x})+\text {proj}_{\vec {f}_2}(\vec {x})+\text {proj}_{\vec {f}_3}(\vec {x})=$ $=\frac {17}{3}\begin {bmatrix}1\\0\\1\\1\end {bmatrix}+\frac {5}{3}\begin {bmatrix}0\\1\\-1\\1\end {bmatrix}+\frac {8}{6}\begin {bmatrix}-2\\0\\1\\1\end {bmatrix}=\frac {1}{3}\begin {bmatrix}9\\5\\16\\26\end {bmatrix}$ $\vec {w}^{\perp }=\vec {x}-\vec {w}=\frac {1}{3}\begin {bmatrix}0\\-2\\-1\\1\end {bmatrix}$ $\vec {x}=\vec {w}+\vec {w}^{\perp }$

Bibliography

Some of the problems come from the end of Chapter 7 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 433–438.

2024-09-06 23:26:22

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Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Solved Problems for Chapter 9

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