(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine if the vectors v_{1}=(3,1,4), v_{2}=(2,-3,5), v_{3}=(5,-2,9), v_{4}=(1,4,-1) span ℝ^{3}

2. Relevant equations

3. The attempt at a solution

So I first arranged it as a matrix,

\begin{bmatrix}

\begin{array}{cccc|c}

3&2&5&1&b_1\\

1&-3&-2&4&b_2\\

4&5&9&-1&b_3

\end{array}

\end{bmatrix}

Now I know what to do if it's a square matrix. I just have to see if the coefficient matrix is invertible (det ≠0). If yes that would mean that any vector b can be expressed as a linear combination. Since this is not a square matrix, I thought I'd have to row reduce it.

Row reduced:

\begin{bmatrix}

\begin{array}{cccc}

1&0&1&1\\

0&1&1&1\\

0&0&0&0

\end{array}

\end{bmatrix}

Now what do I do? It seems to me that the system has infinitely many solutions and therefore the vectors span ℝ^{3}. But the solution manual says that it doesn't. What am I doing wrong? Thanks.

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# Test if these 4 vectors span R^3

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