Can someone help me explain the relationship between two linearly independent vectors and their determinant? Is it always true that the determinant of two lin. ind. vectors is 0? Is there a proof for that?
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An intuitive way of seeing it is that the dimension of the range of a matrix with linearly dependent vectors is always lower than the dimension of the space we're working in. The determinant tells us how much area/volume/etc is scaled, and if we're "going down" one or more dimensions, it will always be $0$. For example, if we're in $\mathbb{R}^3$ and the range of our matrix $A$ is a plane (that is, $\dim (RanA)=2$, volume will be scaled to $0$. I recommend you to watch this https://www.youtube.com/watch?v=Ip3X9LOh2dk. – Douglas Molin Nov 07 '17 at 21:36
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See Using the Determinant to verify Linear Independence, Span, and Basis – wgrenard Nov 07 '17 at 21:36
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You’ve got it backwards: the determinant is nonzero when the rows/columns of the matrix are linearly independent. – amd Nov 07 '17 at 21:42
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@amd so does that mean we cannot infer the determinant if we know they are lin ind? – user7544590 Nov 07 '17 at 22:21
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That is correct. – amd Nov 07 '17 at 23:18