Search for a command to run...
What is Linear Transformation in Matrices. An intuitive explanation with graphs and examples.
I am a developer sharing the problems I solve through my blog. Just giving back to the community I learned so much from! Have an amazing day :)
No comments yet. Be the first to comment.
This series contains the fundamentals of linear algebra. Not so basic but Not so advanced!
What is Matrix Inversion and how to compute them. Explained with examples: inversions for 2x2 and 3x3 matrices
What are Eigenvectors and Eigenvalues. Core concept explained with diagrams, graphs and examples
What are matrix determinants and how to find them. Explained with examples and graphs
What is Matrix Inversion and how to compute them. Explained with examples: inversions for 2x2 and 3x3 matrices
What is Gaussian Elimination and why is it a better choice for solving a system of linear equations (matrices). Explained with examples
On this page
Transformation is another word for function. A function takes in some input, transforms it, and spits out an output. This function or transformation is represented by a matrix. To better understand it, consider the following matrix.
$$A = \left[\begin{array}{} 2 & 0 \\ 2 & 1 \\ \end{array} \right]$$
This matrix represents a certain transformation, to see it in action we can multiply it with an arbitrary vector with coordinates (2, 1)
When we multiply, we get a new vector with coordinates (4, 5) as shown here
$$\left[\begin{array}{} 2 & 0 \\ 2 & 1 \\ \end{array} \right] \left[\begin{array}{} 2 \\ 1 \\ \end{array} \right] = \left[\begin{array}{} 4 \\ 5 \\ \end{array} \right]$$
And when we plot this on the graph we can see how Matrix A "transformed" the vector (2, 1) into another vector (4, 5)
This is why it represents a transformation. But what is so "linear" about it?
For a transformation to be linear it needs to maintain the following two properties after the transformation.
All lines must remain straight
The origin must not move
Considering the example above, we can say that Matrix A can transform any vector in the 2D space into another vector. In other words, the Matrix transforms the entire plane (aka vector space) into a different one while preserving its structure.
A shear matrix is a special matrix that only transforms either one of the two base axes (either X or Y)
Consider a shear matrix transforming a vector with coordinates (0, 2). Numerically we can show the transformation like
$$\left[\begin{array}{} 1 & 1 \\ 0 & 1 \\ \end{array} \right] \times\left[\begin{array}{} 0 \\ 2 \\ \end{array} \right]$$
But when plotted on a graph, it becomes apparent how the transformation reshapes the vector space around it.
You can see in the figure above how a shear matrix transforms a vector (purple arrow) but notice that the transformation also applies to the entire vector space around it (orange arrow and lines).
All the [orange] lines are straight after the transformation
The origin has not moved
Therefore we can say that the shear matrix has "Linearly Transformed" the vector space.
But it isn't always so convenient to imagine/draw a vector space transforming into a new shape. Instead, we focus on the input vectors and just plot how they reshape while keeping the base grid as it is. This helps us focus on how the vectors are affected and not worry about how the entire vector space transforms.
A more generalised way to measure the effect is to apply the linear transformation only to a set of unit vectors - i and j with coordinates (1, 0) and (0, 1) respectively. And the whole grid of the vector space would basically follow the same path
Thank you so much for reading! You can read the next article in the series which explains what are Determinants
If this has helped you in any form, you can show your appreciation by reacting to this article and if you have any questions or feedback please leave them in the comments below
Thanks again!