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

In summary, to prove that the column space of an m x n matrix A is a subspace of R^m, you will need to show that it satisfies the three properties of a subspace: (a) the zero vector of R^m is in the column space, (b) the column space is closed under vector addition, and (c) the column space is closed under multiplication by scalars. Additionally, you will need to show that the column space is a subset of R^m, which can be proven by demonstrating that all the columns of A are from R^m.
  • #1
413
41
0
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
 
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  • #2
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?
 
  • #3
I think i can show the three properties, but how would i show the column space is a subset of R^m?
 
  • #4
413 said:
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 !
 
  • #5
Again, what is the definition of "column space"?
 

What is a column space?

A column space is the set of all possible linear combinations of the columns of a matrix. It represents the space spanned by the columns of the matrix.

How do you prove that the column space of a matrix is a subspace of R^m?

To prove that the column space of a matrix is a subspace of R^m, we need to show that it satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector.

What is the significance of proving that the column space of a matrix is a subspace of R^m?

Proving that the column space of a matrix is a subspace of R^m is important because it provides a fundamental understanding of the properties of vector spaces and their subspaces. It also has practical applications in fields such as linear algebra, physics, and engineering.

Can a matrix have more than one column space?

No, a matrix can only have one column space. This is because the column space is determined by the columns of the matrix, and a matrix cannot have duplicate columns.

What are some common techniques used to prove that the column space of a matrix is a subspace of R^m?

Some common techniques used to prove that the column space of a matrix is a subspace of R^m include using the definition of a subspace, showing that the column space is closed under addition and scalar multiplication, and showing that it contains the zero vector. Other techniques may include using the properties of matrix operations and linear combinations.

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