A First Course in Linear Algebra · Problem LI.M51

Vectors — Robert Beezer

Mathematics — Linear Algebra · spanning-sets-linear-independence

Problem

Consider the subspace W=⟨{v1,v2,v3,v4}⟩W = \langle\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\rangle. Find a set SS so that (1) SS is a subset of WW, (2) SS is linearly independent, and (3) W=⟨S⟩W = \langle S \rangle. Write each vector not included in SS as a linear combination of the vectors that are in SS. v1=[1−12]v2=[4−48]v3=[−32−7]v4=[217]\mathbf{v}_1 = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} \quad \mathbf{v}_2 = \begin{bmatrix} 4 \\ -4 \\ 8 \end{bmatrix} \quad \mathbf{v}_3 = \begin{bmatrix} -3 \\ 2 \\ -7 \end{bmatrix} \quad \mathbf{v}_4 = \begin{bmatrix} 2 \\ 1 \\ 7 \end{bmatrix}
Intuitive explanation

You're finding a basis for the subspace—a minimal set of independent vectors that still spans everything. The key is to identify which vectors are redundant (linear combinations of others) by row-reducing the matrix formed by these vectors as columns. Once you've eliminated the dependent ones, the remaining vectors form your basis SS, and you'll express each discarded vector as a combination of the basis vectors.

Intuitive explanation

You're finding a basis for the subspace—a minimal set of independent vectors that still spans everything. The key is to identify which vectors are redundant (linear combinations of others) by row-reducing the matrix formed by these vectors as columns. Once you've eliminated the dependent ones, the remaining vectors form your basis SS, and you'll express each discarded vector as a combination of the basis vectors.

Related theory

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