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Last Update: 07/19/2023 11:26 PM

Current Deck: Strang, Gilbert; Introduction to Linear Algebra (2016)

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For aa \(3 matrix\times with3\) matrix, 3if columns,we theonly columnslook areat linearlythe independentcombinations ifof
any theytwo arecolumn notvectors, allwe containedwill inget a single two-dimensional plane. TheyThe arekey linearlyquestion dependentis ifwhether theythe arethird allvector containedis in athat singleplane.

Independence: \(w\) is two-dimensionalnot plane.in the plane of \(u\) and \(v\).

Dependence: \(w\) is in the plane of \(u\) and \(v\).
When \(Ax=b\), all linear combinations
\(x_1u + x_2v+x_3w\) lie on the plane given by \(b_1+b_2+b_3=0\).