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I'm an undergrad taking my first linear algebra course, so bear with me.

Whenever we talk about a matrix, is is safe to say that we are discussing a vector space? Vector spaces have very clear definitions and axioms associated with them, but a matrix is just a rectangular layout of numbers. A shorthand, if you will, to make vector spaces easier to deal with. I don't think I've seen matrices used for anything else, but I'm not sure if we can talk about matrices and vector spaces interchangeably or not.

For example, I've been reading Gilbert Strang's "Linear Algebra and its Applications." Let A be a matrix. It makes sense to talk about the column space of A, or the null space of A, or the row space of A. But does it make sense to talk about the dimension of A or the basis of A?

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  • $\begingroup$ Well you can view the row/column vectors as a basis of a vector space - so if you cut out linear dependent vectors you will end up with a basis that spans a vector space - for those vectors you have the same meaning of a dimension e.g. $\endgroup$ Dec 2 '17 at 22:02
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No. An $(n \times n)$ matrix represents a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^n$, not a vector space.

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  • $\begingroup$ That is when we are considering set of matrices not one matrix. $\endgroup$
    – Mr. X
    Dec 2 '17 at 22:08
  • $\begingroup$ I thought a linear transformation was a vector space? $\endgroup$ Dec 2 '17 at 22:09
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The column space is the image of a matrix. What I mean, is take a basis $\{e_1, \dots e_n\}$ the vectors $Ae_1, \dots A e_{n}$ are each vectors in their own right (in fact, you should check that if $A$ is written with respect to the basis, then these are precisely the column vectors of a matrix.) The image of a linear transformation is a subspace of the codomain, which is what is meant by "column space."

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  • $\begingroup$ So I guess what you're saying is that any matrix CAN represent a vector space, but it doesn't have to? (Because like you said, a matrix is composed of vectors, and those vectors can form a basis for a vector space) $\endgroup$ Dec 2 '17 at 22:14
  • $\begingroup$ No. I’m sayin a matrix sends the basis vectors to a vector space known as the image of the matrix. The collection of columns span the image. It’s somewhat subtle $\endgroup$ Dec 2 '17 at 22:15

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