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'm confused as to the definition of column space. Looking at this exercise, my understanding is that column space is the vectors that make up the span of the matrix? Could you maybe clear up the definition of column space and how it applies to this example?

'm confused as to the definition of column space. Looking at this exercise, my understanding is that column space is the vectors that make up the span of the matrix? Could you maybe clear up the definition of column space and how it applies to this example?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I'm confused as to the definition of column space. Looking at this exercise, my understanding is that column space is the vectors that make up the span of the matrix? Could you maybe clear up the definition of column space and how it applies to this example?

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Step 1

Let A∈Fm×n where F is a field. We have seen N(A), the nullspace of A, is given by {x∈Fn:Ax=0}

The column space of A, denoted by C(A), is the span of the columns of A. In other words, the we treat the columns of A as vectors in Fm and take all possible linear combinations of these vectors to form the span. So C(A) is a subspace of Fm.

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