[Math] Show that set of all $2 \times 2$ matrices forms a vector space of dimension $4$

linear algebramatricesvector-spaces

I have this question:

Show that the set of all $2 \times 2$ matrices with real coefficients forms a linear space over $\Bbb R$ of dimension $4$.

I know that the set of the matrices will basically form a linear combination which will define the vector space and they satisfy the axioms defined for the vector space.

I do not know how to show that this is possible. Do the vectors have to be linearly independent?

Any sort of help is appreciated.

Thanks.

Best Answer

Begin by listing the axiom satisfied by a general vector space over $\mathbb{R}$. Now consider your set of matrices. What does you addition look like? What is your zero element? What does your scalar multiplication look like? Does your scalar multiplication and your addition behave as they should? Finally, show that all 2 by 2 matrices can be written as a linear combination of 4 special matrices. A good choice is to pick matrices who each have precisely 3 zero entries and 1 non zero entry. Can you show that these 4 matrices are linearly independent?

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