In a linear algebra book one problem is as follows: "given a linear map from $\Bbb R^2$ to $\Bbb R^2$ defined in terms of standard basis the problem is to find the image of $(x,y)$. "
The solution I know is to express $(x,y)$ as a linear combination of standard bases and then use linearity of the map and then put the values of the images of stanard bases.
In the book I see the following solution : represent the map in matrix form with respect to the standard bases and multiply it with the vector (x,y) represented as column vector. I could not understand how these two are equivalent. How is matrix multiplication coming into picture. I am aware of the fact that if $T$ and $S$ are two linear maps then wrt to a basis the matrix representation is same as the multiplication of the two matrices for each of the linear maps wrt to the same basis