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Multiplying Vector by a Matrix

As we start to learn more about matrices, we may be asked to multiply a vector by a matrix. If you're not quite sure how to do this, you're in the right place. As we'll soon discover, there are a few important things to consider when it comes to carrying out multiplication operations with matrices. But remember: The whole point of matrices is to simplify algebraic operations into relatively simple arithmetic operations. Once we understand how this process works, we can expect matrices to make our lives easier. Let's find out more:

What is a vector?

In mathematics, the term vector has a distinct meaning. In physics and geometry, a vector typically refers to a quantity that has both direction and magnitude, such as velocity. In the context of linear algebra and matrices, a vector takes on a slightly different meaning.

A vector is a list of numbers, arranged either in a row (referred to as a row vector) or a column (known as a column vector). For example, these are both vectors:

Row vector: $\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$

Column vector:

Vectors can be considered as special kinds of matrices. Specifically, a vector is a matrix with only one row or one column. In this sense, every vector can be viewed as a matrix, but not every matrix is a vector.

Furthermore, matrices can contain multiple vectors within them. For example, each row or column in the following matrix is a vector:

In this $3×3$ matrix, there are three row vectors and three column vectors.

It's crucial to understand the distinction and relationship between vectors and matrices, as they are fundamental to many areas of mathematics, data analysis, machine learning, computer graphics, and more.

Multiplying vectors by matrices

There are a few rules we need to remember before we start multiplying vectors by matrices:

• In order to multiply a matrix by a vector, the matrix must have as many columns as the vector has rows

According to this rule, this operation would not be possible:

Why? Because the variable matrix does not have as many columns as the coefficient matrix.

However, this operation would be possible:

We can see that everything lines up, and it's clear that we can move ahead with our vector multiplication in this case.

Now consider this multiplication:

What is $AY$ if:

And

To start, we would line the vector up with the matrix:

Our first step is to multiply row 1 of our matrix by column 1 of the vector. Note that there is only one column in our vector, so we are simply multiplying the first row of our matrix by the entire vector. Let's line them up:

Before we go any further, we need to understand one very important fact:

Multiplying operations with matrices is not commutative. In other words, we do not use the FOIL method as we would when multiplying numerous terms in two different brackets. We only have to multiply each of the two numbers once. This makes more sense when we write out this operation:

Now we can take all of these products and write them in a special vector called a "matrix-vector product." Here's what that looks like:

Why would we want to multiply a vector by a matrix?

There are several reasons why we might want to multiply a vector by a matrix. One possible reason is to write our matrix as a function. We might also find ourselves multiplying matrices by vectors when solving linear equations. As you may recall, using matrices to solve linear equations is one of the key applications of matrices in general. Perhaps a more obvious reason is simple: Multiplying a vector by a matrix is much simpler than multiplying two different matrices. In other words, it helps us understand the basics of scalar multiplication before moving on to more challenging concepts.

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