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Math / Linear Algebra: A Geometric Approach 2 / Chapter 3.4 / Problem 13

Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams

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Let A be an m n matrix with rank r. Suppose A = BU, where U is in echelon form. Show

Chapter 3, Problem 13

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QUESTION:

Let A be an m n matrix with rank r. Suppose A = BU, where U is in echelon form. Show that the first r columns of B give a basis for C(A). (In particular, if EA = U, where U is the echelon form of A and E is the product of elementary matrices by which we reduce A to U, then the first r columns of E 1 give a basis for C(A).)

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QUESTION:

Let A be an m n matrix with rank r. Suppose A = BU, where U is in echelon form. Show that the first r columns of B give a basis for C(A). (In particular, if EA = U, where U is the echelon form of A and E is the product of elementary matrices by which we reduce A to U, then the first r columns of E 1 give a basis for C(A).)

ANSWER:

Step 1 of 2

It is given that, is an matrix with rank .

And,

.

Also, is in echelon form.

It is known that,

If be a -dimensional subspace. Then any vectors that span must be linearly independent and any linearly independent vectors in must span .

To prove that, the first columns of give a basis for .

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