For the subspace below, (a) find a basis for the subspace, and (b) state the dimension. {[a b c d] : a - 2b + 5c = 0} (a) Find a basis for the subspace. A basis for the subspace is { }. (Use a comma to separate matrices as needed.) (b) State the dimension. The dimension is
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Step 1:** Consider the vectors given in the explanation: \[ \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 5 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \] ** Show more…
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For the subspace below, (a) find a basis, and (b) state the dimension. a. Find a basis for the subspace. A basis for the subspace is { }. (Use a comma to separate vectors as needed.) b. State the dimension. The dimension is .
Let $A=\left[\begin{array}{rrrr}1 & 1 & -1 & 1 \\ 2 & -3 & 5 & -6 \\ 5 & 0 & 2 & -3\end{array}\right],$ and let $\mathbf{v}_{1}=$ (-2,7,5,0) and $\mathbf{v}_{2}=(3,-8,0,5)$. (a) Show that $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ is a basis for the null space of $A$. (b) Using the basis in part (a), write an expression for an arbitrary vector $(x, y, z, w)$ in the null space of $A$.
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