Simple Matrix Multiplication and Vector Multiplication

Matrix multiplication, vectors, linear spaces have sometimes horrifying effects on students trying to understand them. And the lectures on these topics offer very little intuitive fundaments on which the student can build their knowledge upon. In this article I will be giving you a little intuitive information on what are matrices, and how can we apply matrix operations.

Matrixmultiplikation, Vektoren, lineare Räume waren immer noch schreckliche Dinge für Studenten. Die Inhalten der Unterrichten bieten jedoch keine Intuitiven, damit die Studenten wirklich verstehen kann, was Matrices und linear Räume sind. In diesem Artikel werde ich Ihnen ein paar Informationen über Matrizen und lineare Räume geben, damit Sie die geometrische Bedeutungen von Matrixoperationen verstehen können.

Before we begin I would like to recommend a playlist from a YouTube channel which I find is extremely successful in painting the geometric meanings of linear algebra, 3Blue1Brown‘s Essence of Linear Algebra Series. You can also check out my other posts about Creative Arts, Engineering Mathematics, Engineering Mechanics.

What is actually a Linear Space?

When we think of coordinates and coordinate systems, more often than not we will think about this basic shape of 3 arrow , each one perpendicular to the others.

This representation of base vectors is chosen as the standard base with each base vector being 1 unit long. In this depiction the base vectors have coordinates given in terms of standard base vectors. They are:

x=x_1 = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} , y= x_2= \begin{pmatrix}0\\1 \\0 \end{pmatrix}, z=x_3= \begin{pmatrix}0\\0 \\1 \end{pmatrix}

What would happen if we squished and stretched the base vectors around and the base vector x_1 ended in \begin{pmatrix}3\\0 \\0 \end{pmatrix} instead?

This is what the mathematicians call a Transformation. It transforms the Vectorspace from as we know it to an arbitrary layout of base vectors.

What is a linear transformation?

When we construct a grid on the standard depiction of 3d Coordinates, this Cartesian coordinate system is the result.

We can see the parallel lines constructed alongside the x_1 and x_2 axis. In a linear transformation these lines remain straight, and parallel to each other.

What happens after a transformation?

Transformations are a mere different view on the same topic. They span the same space with different vectors. Therefore the coordinates of a point in space changes. Example vector before and after the transformation

x_1 = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} \rightarrow x_1 = \begin{pmatrix} 3\\ 0\\ 0 \end{pmatrix}

The coordinates of Point A = \begin{pmatrix} 3\ 0\ 0 \end{pmatrix} will be \begin{pmatrix} 1\ 0\ 0 \end{pmatrix} The new coordinates of the Point can be easily calculated with the help of linear combination and gaussian elimination.

Basically we take the new base vectors and linearly combine them to get to the Point A = \begin{pmatrix} 3\ 0\ 0 \end{pmatrix} . The Coefficients of linear combination will give us the new coordinates of A. See gaussian elimination post.

So what does now a matrix represent?

A matrix consists of rows and columns, perfect for vector storage. We take our base vectors x_1, . . . , x_n and write down their new end points, respectively, in a matrix.

For example our transformation matrix \mathbb{L} would look like this:

\mathbb{L} = \begin{pmatrix} 3 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 &1 \end{pmatrix}

So matrices will represent the transformation the space endures. But how can we execute Vector Matrix Multiplications?

Vector Matrix Multiplication

Now think of Vector matrix multiplications as applying a vector the transformation of its space.

When we squish and stretch the base vectors of a space we also squish and stretch the base vectors of a vector in that space. After the transformation the coordinates (1,1,1) does not end in the same spot it did before the transformation.

To find where the new vector ends we apply the transformation matrix \mathbb{L} to the Vector v. The resulting vector is where the original vector lands after the transformation.

v= \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} , v_{transformed}= \begin{pmatrix} 3 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 &1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}

= \begin{pmatrix} ^{x_1}X_1 & ^{x_2} X_1 & ^{x_3} X_1 \\ \;\;X_2 & \;\;\, X_2 & \;\; X_2 \\ \;\; X_3 & \;\; X_3 & \;\; X_3 \end{pmatrix} \cdot \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}

Each column represents the end point of each base vector with respect to standard base vectors.

Then the multiplication can be easily made:

v_1 \begin{pmatrix}^{x_1}X_1 \\ X_2 \\ X_3\end{pmatrix} + v_2 \begin{pmatrix}^{x_2}X_1 \\ X_2 \\ X_3\end{pmatrix} + v_3 \begin{pmatrix}^{x_3}X_1 \\ X_2 \\ X_3\end{pmatrix} = v_{transformed}

Example :

\begin{pmatrix} 3 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 &1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = 1 \begin{pmatrix}3\\ 0\\ 0\end{pmatrix}+2 \begin{pmatrix}0\\ 1\\ 0\end{pmatrix} + 3 \begin{pmatrix}0\\ 0\\ 1\end{pmatrix} = \begin{pmatrix}3\\ 2\\ 3\end{pmatrix}

We can control this calculation on Casio‘s Calculator Page and then we can transition to matrix-matrix multiplication.

Matrix Matrix Multiplication?

The same thinking applies in matrix matrix multiplication too. This time we apply more than one transformation to the space.

We have talked about imagining the transformation matrix as a summary of vectors representing where the former base vectors will end after the transformation.

When we have two matrices ,for example, we can separate the one on the right side mentally into base vectors, also its columns. Now execute Vector Matrix multiplications for each of the columns, and write each solution vector the the corresponding column in a new matrix.

The resulting matrix is our total transformation matrix from two transformation matrices.