Множення матриці на вектор

A vector is just a matrix that is $1\times n$ or $m\times1$. So in order to multiply you just need to do the same thing as when you multiply two matrices together.

Example

$$\begin{bmatrix}1&2\\3&4\end{bmatrix}*\begin{bmatrix}5\\10\end{bmatrix}=\begin{bmatrix}25\\55\end{bmatrix}$$ $$\begin{bmatrix}5&10\end{bmatrix}*\begin{bmatrix}1&2\\3&4\end{bmatrix}=\begin{bmatrix}35&50\end{bmatrix}$$

Notice how the product is always in vector form.

Multiplying a $2\times2$ matrix by a $2\times1$ vector will ALWAYS give you a $2\times1$ vector.

Example

Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$

Let $\vec{v}=\begin{bmatrix}e\\f\end{bmatrix}$

$A*\vec{v}=\begin{bmatrix}a&b\\c&d\end{bmatrix}*\begin{bmatrix}e\\f\end{bmatrix}=\begin{bmatrix}ae+bf\\ce+df\end{bmatrix}$

Notice that $A\vec{v}$ is $2\times1$.

(Video) Multiply a Matrix and Vector

by giant_neural_network

This video by giant_neural_network teaches you how to multiply a matrix and vector together.

Summary

We're given a matrix $\begin{bmatrix} 2&-1 \\ 1&1 \\ \end{bmatrix}$ and a vector $<1,2>$, which we can also write as $\begin{bmatrix} 1 \\ 2 \\ \end{bmatrix}$. Matrices with one column are more commonly known as vectors.

To multiply a matrix and a vector, we take the numbers in the vector and line them up with the column of the matrix. Then, we consider the columns of the matrix as vectors themselves, splitting the matrices into vectors. Then, we will add the new vectors multiplied by the indiivdual components of the original vector Essentially, what we are aiming for is:

$$1 \cdot \begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} + 2 \cdot \begin{bmatrix} -1 \\ 1 \\ \end{bmatrix}$$

We will use scalar multiplication to find the solution.

$$\begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} + \begin{bmatrix} -2 \\ 2 \\ \end{bmatrix}$$

We can now add these matrices together.

$$\begin{bmatrix} 0 \\ 3 \\ \end{bmatrix}$$

That's our final answer!

$$\begin{bmatrix} 2&-1 \\ 1&1 \\ \end{bmatrix} + <1,2> = \begin{bmatrix} 0 \\ 3 \\ \end{bmatrix}$$

Multiplying a Matrix by a Vector

Now that we've learned the basics of the row and column matrix, we can move forward into more complicated types of matrix multiplication.

In this next lesson in Matrix Multiplication, we will learn how to multiply a matrix by a vector. Matrix–matrix multiplication extends the ideas used when multiplying a matrix by a vector. It provides a compact way to represent a system of linear equations.

This concept is often confused with finding the dot product of vectors. Don't fall into this trap!

Before multiplying matrices, you must determine whether the product matrix is defined and what its dimensions will be.

In the above image, the number of rows is $2$, and the number of columns is $2$. Vectors can be re-written as column matrices. This concept is essentially vector multiplication, but with matrices.

Reflecting a figure involves multiplying a reflection matrix by each point’s vector.