- #1
jey1234
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Homework Statement
Determine if the vectors v1=(3,1,4), v2=(2,-3,5), v3=(5,-2,9), v4=(1,4,-1) span ℝ3
Homework Equations
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 ℝ3. But the solution manual says that it doesn't. What am I doing wrong? Thanks.