Proving Theorem: Column Space of Matrix A is a Subspace of R^m

[413]

How would I prove this theorem:

"The column space of an m x n matrix A is a subspace of R^m"

by using this definition: A subspace of a vector space V is a subset H of V that has three properties: a) the zero vector of V is in H. b) H is closed under vector addition. c) H is closed under multiplication by scalars.

Please help

 

[HallsofIvy]

Okay, here's what you should do:
1. Write out the definition of "column space" since you already have the definition of "subspace".

2. Show that the column space is a subset of R^m.

3. Show that (a) is true: is the zero vector in the column space- does it satisfy the definition of vectors in the column space?

4. Show that (b) is true: if you add two vectors in the column space is the result in the column space?

5. Show that (c) is true: if you multiply a vector in the column space by a scalar is the result in the column space?

 

[413]

I think i can show the three properties, but how would i show the column space is a subset of R^m?

 

[radou]

I think i can show the three properties, but how would i show the column space is a subset of R^m?

Well, it's obvious since all the columns are from R^m !

 

[HallsofIvy]

Again, what is the definition of "column space"?