Linear Algebra: Intuition, Math, Code by Cohen Mike X

Linear Algebra: Intuition, Math, Code by Cohen Mike X

Author:Cohen, Mike X [Cohen, Mike X]
Language: eng
Format: epub
Publisher: Sincxpress BV
Published: 2021-02-03T16:00:00+00:00


Figure 10.6: A graph of the column spaces before and after RREF for matrix M (Equation 10.30 ). Column vectors are drawn on top of the planes (unit-normalized for visualization).

On the other hand, keep in mind that row reduction is not necessarily guaranteed to change the column space. Consider the following matrix and its RREF:

It’s a full-rank matrix and its RREF is the identity matrix. The column space spans all of ℝ2 both before and after row reduction. Now this situation is analogous to the row space: The elements in the columns are different but the subspaces spanned by those columns are exactly the same.

Again, the take-home messages in this section are that row reduction (1) does not affect the rank or the dimensionalities of the matrix subspaces, (2) does not change the row space, and (3) can (but does not necessarily) change the column space.



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