Prove W is a Subspace of m2,2 - Linear Alg.

[sportlover36]

so on a homework i got a problem that asks this.
Let W be a set of 2x2 matrix with a form of

|a , 0|
|a+c , c|

Prove that W is a subspace of m2,2

 

[HallsofIvy]

By using the definition of "subspace", of course! To show that "X" is something, you show that X satisfies the definition of that something. Here we can simplify a lot. The definition of "subspace" is that it is a subset of a vector space that can be considered a vector space itself. That means we can take things like the existence of the commutativity of addition, associativity of addition, etc. as given since they are true for anything in the original space and so true for the subset. Basically what you must prove is that the subset is closed under addition and scalar multiplication. You must also prove that the set is non-empty. Often that is done by showing that the 0 vector is in the set.

Here, W is the set of matrices of the form
\begin{bmatrix}a & 0 \\a+c & c\end{bmatrix}

What would a and c have to be to give the 0 matrix?

If you add two matrices of thaf form is the result also of that form?
Look at
\begin{bmatrix}a & 0 \\a+c & c\end{bmatrix}+ \begin{bmatrix}b & 0 \\b+ d & d\end{bmatrix}
is that sum of the same form?

What about scalar multiplication. If b is a number, is
b\begin{bmatrix}a & 0 \\a+c & c\end{bmatrix}
of that form?

 

[Dick]

You prove it's a subspace by using the definition of a subspace. Can you look that up and tell us what it is?