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nitraiddQ

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2021-11-05

Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?
$[\begin{matrix} 1 \\ - 3 \\ 2 \\ - 4 \end{matrix}], [\begin{matrix} - 3 \\ 9 \\ - 6 \\ 12 \end{matrix}], [\begin{matrix} 2 \\ - 1 \\ 4 \\ - 2 \end{matrix}], [\begin{matrix} - 4 \\ 5 \\ - 3 \\ 7 \end{matrix}]$

Answer & Explanation

smallq9

Skilled2021-11-06Added 106 answers

The subspace covered by the provided vectors and its dimension need to have a basis.
The augmented matrix
$[\begin{matrix} 1 \\ - 3 \\ 2 \\ - 4 \end{matrix}], [\begin{matrix} - 3 \\ 9 \\ - 6 \\ 12 \end{matrix}], [\begin{matrix} 2 \\ - 1 \\ 4 \\ 2 \end{matrix}], [\begin{matrix} - 4 \\ 5 \\ - 3 \\ 7 \end{matrix}]$
Reduce matrix to row echelon form
$[\begin{matrix} 1 \\ 0 \\ 0 \\ - 4 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 0 \\ 12 \end{matrix}], [\begin{matrix} 2 \\ 5 \\ 0 \\ 2 \end{matrix}], [\begin{matrix} - 4 \\ 7 \\ 5 \\ 7 \end{matrix}] \begin{matrix} R_{4} - 4 R_{1} \to R_{4} \\ R_{4} - 2 R_{2} \to R_{4} \end{matrix} \sim [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 5 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 4 \\ 7 \\ 5 \\ 5 \end{matrix}]$
$[\begin{matrix} 1 \\ 0 \\ 0 \\ - 4 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 0 \\ 12 \end{matrix}], [\begin{matrix} 2 \\ 5 \\ 0 \\ 2 \end{matrix}], [\begin{matrix} - 4 \\ 7 \\ 5 \\ 7 \end{matrix}] \begin{matrix} R_{4} - 4 R_{1} \to R_{4} \\ R_{4} - 2 R_{2} \to R_{4} \end{matrix} \sim [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 5 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 4 \\ 7 \\ 5 \\ 5 \end{matrix}]$
$[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 5 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 4 \\ 7 \\ 5 \\ 5 \end{matrix}] R_{4} - R_{3} \to R_{4} \sim [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 5 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 4 \\ 7 \\ 5 \\ 5 \end{matrix}]$
As a result, the first, third, and fourth pivot columns in the matrix are revealed. This is the foundation for the subspace that the provided vector spans.
$[\begin{matrix} 1 \\ - 3 \\ 2 \\ - 4 \end{matrix}], [\begin{matrix} 2 \\ - 1 \\ 4 \\ 2 \end{matrix}], [\begin{matrix} - 4 \\ 5 \\ - 3 \\ 7 \end{matrix}]$
and $\dim H = 3$

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