Multiplying a Vector by a Matrix
To multiply a row vector by a column vector, the row vector must have as many columns as the column vector has rows.
Let us define the multiplication between a matrix and a vector in which the number of columns in equals the number of rows in .
So, if is an matrix, then the product is defined for column vectors . If we let , then is an column vector. In other words, the number of rows in determines the number of rows in the product .
The general formula for a matrix-vector product is
Example :
Find where and .
By the definition, number of columns in equals the number of rows in .
First, multiply Row of the matrix by Column of the vector.
Next, multiply Row of the matrix by Column of the vector.
Finally multiply Row of the matrix by Column of the vector.
Writing the matrix-vector product, we get: