MAT $242,$ Fall $2024$

Written Homework $4$

Instructor: Oleksandr Lytvak

$1.(5$ points$)$ Let B $=\{$~v$1,$ ~v$2,$ ~v$3\}$ be the set of vectors, where ~v$1=$

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$3$

$4$

$1$

$4$

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$,$ ~v$2=$

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$$

$5$

$2$

$3$

k

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$$

$,$

~v$3=$

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$$

$2$

m

$2$

$8$

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$.$ Find the values for k and m such that the set B is orthogonal.

$2.(5$ points$)$ Let W $=$span$\{$~v$1,$ ~v$2\}$ where ~v$1=$

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$1$

$3$

$5$

$6$

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$,$ and ~v$2=$

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$2$

$1$

$4$

$7$

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$.$ Find a basis for

W $,$ the orthogonal complement to W $.$

$3.(15$ points$)$ Let W $=$span$\{$~v$1,$ ~v$2,$ ~v$3\},$ where ~v$1=$

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$1$

$2$

$0$

$2$

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$$

$,$ ~v$2=$

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$0$

$6$

$4$

$6$

$$

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$,$ and ~v$3=$

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$3$

$5$

$1$

$7$

$$

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$.$

$($a$)$ Show that the set of vectors B $=\{$~v$1,$ ~v$2,$ ~v$3\}$ forms a basis for W $.$

$($b$)$ Determine whether the set of vectors B is orthogonal. Show your work.

$($c$)$ Find the orthogonal projection of vector ~u $=$

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$5$

$2$

$4$

$3$

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$$ onto W $.$

$($d$)$ Use your answer for part $($c$)$ to determine whether the vector ~u is in W $.$