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$[\mathbf{M}]$ Find a column of the matrix in Exercise 39 that can be deleted and yet have the remaining matrix columns still span $\mathbb{R}^{4} .$

since $B^{o}$ is in echelon form, it shows that $A^{o}$ has a pivot position in each row. Therefore, the columns of $A^{o} \operatorname{span} R^{4} .$ It is possible to delete column 3 of $\mathrm{A}$ instead of column $4 .$ In this case, the fourth column of A becomes a pivot column of $A^{o},$ as you can see by looking at what happens when column 3 of $\mathrm{B}$ is deleted.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 4

The Matrix Equation Ax D b

Introduction to Matrices

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

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02:45

$[\mathbf{M}]$ Find a colu…

05:21

[M] Let $A$ be the matrix …

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[M] In Exercises $37-40$ ,…

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The ________ $M_{ij}$ of t…

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Find (a) $\operatorname{de…

03:19

Suppose a $4 \times 7$ mat…

02:59

Determine the null space o…

for this example were considering a matrix A that's displayed here and using a computer algebra system. We haven't row reduced to its role, reduce echelon form. Where comes 123 and five are Pivot Collins. Here we find that all rows of a are pivot rose, and this means the following Since every row of a contains a pivot, this means that the span of the columns of A is our four itself. So take each one of these column vectors inside the Matrix. A. If you spend this set, you'll find the set is identically are for Well, this is interesting because The Matrix A has five columns, and all we needed was a pivot and every row in order to spend all of our four. So now the question becomes, Could we delete a column and still have a pivot in every row or another? White. In other words, which column can we delete so that a still spans all of our four? To answer such a question, we have to go to the non pivot column. So this is not a pivot column and notice that if we did delete this column outright so that we have a new matrix. Let's call it a hat where this column is missing. A hat still has, ah, pivot in every column, and so a hat still spans that target space. All of our four. So deleting the fourth column has no effect on what the span will be since we still have a pivot in every row.

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