matrix multiplication row X square

Let's begin with a very simple multiplication of two matrices, a row matrix times a square matrix. Notice that number of rows/columns of the square matrices (three by three in the first example and four by four in the second) is equal to the number of elements in the row matrices they are being multiplied by. In graphics this the the common situation where you are multiplying the row matrix for a point by a square transformation matrix.
First look at the left hand 3 by 3 example. Note that the result is a three element row the same size as the original row matrix. Just as a three element row matrix produces a three element row matrix answer, so a four element row matrix in the second example produces a four element row matrix answer. Notice how the matrix multiplication is performed. The elements of the row matrix are multiplied by the elements of each column and added together, one column at a time. This gives you a number for each column, and these numbers represent the elements of the resulting row matrix.

matrix multiplication square X column (post multiply)
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