The multiplication of a matrix vector

In this blog, I'm going to deal with the multiplication of a matrix vector.

 

1. A matrix : it's just an array of numbers. If i have an m by n matrix, the m is the number of rows, and the n is the number of columns.

2. Taking the product of matirx A with some vector x : It only works if the vecotr, x, has the same number of components as A has columns. In other words, if the length of vector x, or the number of components in vector is equal to the number of columns in matrix A, then we define A times our vector x.

3. a column vector and a row vector : when you say a column vector, it means the numbers in a vecotr is arranged in vertical direction. But a row vector is literally the numbers arranged in horizontal direction. Although those vectors have different forms, we can turn one vector form into the other vector form by using the transpose function. In transpose, you turn the rows into columns and the columns into rows.

With this concept, we can view the multiplication as a group of the dot products of each row vector and vector x. It's what i said.

And there is another way to view this situation. The first way is to take the transpose functions of column vectors, and then take the dot products of the changed vectors and a vector out of the matrix. But in the second way, we consider a matrix as a set of column vectors. It sounds a bit abstract. Here there is.

There were 12 entries in the matrix A. But I turned them into 4 column vectors. And now i can do the multiplication of matrix A and vector x.

as the definition of the multiplication, we can rewrite the matrix as the linear combination of the 4 column vectors.