Proving Real 2x2 Matrices are a Vector Space

[dmitriylm]

Homework Statement

Show that all 2 x 2 matrices with real entries:

M(2x2) = {
a b | a,b,c,d are real numbers}
c d |

is a vector space under the matrix addition:

|a1 b1| + | a2 b2| = |a1+a2 b1+b2|
|c1 d1| + | c2 d2| = |c1+c2 d1+d2|

and scalar multiplication:

r*| a b | = | ra rb |
r*| c d | = | rc rd |

This vector space is "similar" to another vector space. Can you comment on this?

*Ignore any silver text, its only used for formatting.

How would I go about proving this?

 

[dmitriylm]

I think I've actually got this figured out. I just test two 2x2 matrices of random values under the various axioms and if they pass then under matrix addition and scalar multiplication then it should show that all 2x2 matrices are a real vector space right?

What other kind of vector space is this similar to?

 

[Mark44]

I think I've actually got this figured out. I just test two 2x2 matrices of random values under the various axioms and if they pass then under matrix addition and scalar multiplication then it should show that all 2x2 matrices are a real vector space right?

What other kind of vector space is this similar to?

By "random" values, I think you really mean arbitrary values - in other words, unspecified values. Yes, that's the right approach. You'll need to verify all ten of the axioms (or 11 or whatever).

Each 2 x 2 matrix has four entries. Can you think of another vector space whose vectors have four values? I think this is where the book is leading you.

 

[dmitriylm]

Would that be a single column matrix like:

[a]

[c]
[d]

?

 

[Mark44]

Yes, that's where I think they're leading you.