So, beforehand i need to say that i have the solution for this problem given by my teacher, but i dont really understand why it works. Here is the problem:
Given the set
\begin{pmatrix} 1&0 \\ 0&1 \\ \end{pmatrix} ,\begin{pmatrix} 1&0 \\ 0&-1 \\ \end{pmatrix},\begin{pmatrix} 0&1 \\ 0&0 \\ \end{pmatrix},\begin{pmatrix} 0&0 \\ 1&0 \\ \end{pmatrix}
Find an Inner product so that the elements in the set form an orthonormal basis of M2.
I know that the cross product between the elements of the basis must be 0 and the norm must be 1, but i dont know how the process works to construct an inner product that fits the needs of the problem.
Here is the link to the solution(i cant link images yet, i apologize), i dont really get what is done. I understand that the product he created works and the basis is indeed orthonormal, but i dont understand why it works. How, in that process, is it guaranteed that the product he gets will result in the product between basis being 0 and the norm being 1.
In a vector space $V$ we can give an inner product such that a chosen basis $\{v_1, v_2, v_3, v_4\}$ is orthormal simply by defining $$\left\langle \sum \limits_{i=1}^4 \alpha_iv_i, \sum \limits_{i=1}^4 \beta_iv_i\right \rangle = \sum \limits_{i=1}^4\alpha_i\beta_i.$$ You can convince yourself of this by checking orthonomality conditions.
In the solutions they first are figuring out how any vector (matrix) can be written as a linear combination of the 4 basis vectors given. This gives the coefficients $\gamma, \mu, \ldots$ in terms of the entries of the original matrix $a,b, \ldots$. Then they are using the above fact to replace the coefficients of the basis expansion with the entries of the matrix to give an explicit formula for the inner product.