How do I write a vector as a linear combination of other vectors.

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Write $\begin{pmatrix} 5 \\ 3 \\15 \end{pmatrix}$ as a linerar combination of the following vectors: $u=\begin{pmatrix} 1 \\ 2 \\5 \end{pmatrix}$, $v=\begin{pmatrix} 3 \\ -4 \\-1 \end{pmatrix}$, $w=\begin{pmatrix} -1 \\ 1 \\1 \end{pmatrix}$.

My attempt:

$$\begin{bmatrix} 1 & 3& -1 & 5\\ 2 & -4 & 1& 3\\ 5&-1&1&15\\ \end{bmatrix}\sim\to\begin{bmatrix} 1 & 0& 0 & 3\\ 0 & 1 & 0 & 1\\ 0&0&1&1\\ \end{bmatrix}$$

Obviously I skipped a lot of reduction steps, because it's a pain to type matrices here, but I was wondering if I even did the right thing.

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2 Answers

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So...you've done the hard part, now finish it off:

$$c_1\begin{pmatrix} 1 \\ 2 \\5 \end{pmatrix} + c_2 \begin{pmatrix} 3 \\ -4 \\-1 \end{pmatrix}+ c_3\begin{pmatrix} -1 \\ 1 \\1 \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \\15 \end{pmatrix}$$

So what have you found to be $c_1, c_2, c_3$?

Fill in the correct values for each constant, and you're done!

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You can set this problem up as linear system of 3 equations

$$ \begin{pmatrix} 5 \\ 3 \\ 15 \end{pmatrix} = \begin{bmatrix} 1 & 3 &-1 \\ 2 & -4 & 1 \\ 5 & -1 & 1 \end{bmatrix} \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} $$

You do this by arranging the basis vectors as columns to the coefficient matrix.

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