A First Course in Linear Algebra · Problem LI.M50

Vectors — Robert Beezer

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

Problem

Consider the set of vectors from C3\mathbb{C}^3, WW, given below. Find a set TT that contains three vectors from WW and such that W=⟨T⟩W = \langle T \rangle. W=⟨{v1,v2,v3,v4,v5}⟩=⟨{[211],[−1−11],[123],[313],[01−3]}⟩W = \langle\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4, \mathbf{v}_5\}\rangle = \left\langle \left\{ \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ -3 \end{bmatrix} \right\} \right\rangle
Intuitive explanation

The key insight is that WW is a subspace spanned by five vectors, but it likely has dimension less than 5—meaning some vectors are redundant. To find a minimal spanning set TT of exactly three vectors, use row reduction: form a matrix with these vectors as columns, reduce to row echelon form, and identify which original columns correspond to pivot positions. Those pivot columns are linearly independent and span the same space as all five vectors.

Intuitive explanation

The key insight is that WW is a subspace spanned by five vectors, but it likely has dimension less than 5—meaning some vectors are redundant. To find a minimal spanning set TT of exactly three vectors, use row reduction: form a matrix with these vectors as columns, reduce to row echelon form, and identify which original columns correspond to pivot positions. Those pivot columns are linearly independent and span the same space as all five vectors.

Related theory

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