In Week3, Linear Transformation part, Lecture “Matrices as linear transformations” at minute 0:55 -
The professor says, those two coordinates (i.e. a,b) represent a column vector. Shouldn’t that be row vector instead?
If you look at the matrix in the picture, which I try to represent here:
a b
|3 1|
|1 2|
The point (3,1) where a=3 and b=1, forms the row vector on row1 (or row0 if its 0 index to you), isn’t it?
Answering my own question here.
By saying “those two coordinates represent a column vector” reference to the vector being multiplied to the matrix. I understand it as a transpose and hence the coordinates also get transposed from the matrix. This makes sense.
But now I have the question on what the matrix represents? Is it a set of equations, each as a row vector stacked (this is what caused the confusion in my head, the topic of the question)? And does that mean that we are using a system of equations here for that matrix?
The matrix represents a linear transformation from \mathbb{R}^2 to \mathbb{R}^2. We can express that as two linear equations or as a single matrix multiply with the input vector as a column vector. The equations would be:
ax + by = e
cx + dy = f
Or as the matrix multiply, you can express the same transformation as:
A = [[a, b], [c, d]]
v_{out} = A \cdot [[x], [y]]
Hello @parth105,
I would like to expand from @paulinpaloalto’s answer to cover a bit more.
When I learned the topic of “Linear transformation and Matrices”, I found these intuitive, geometric examples very helpful.
Take “rotation” as an example and set \theta=90^{\circ}=\frac{\pi}{2}: \begin{bmatrix} \cos\frac{\pi}{2} & -\sin\frac{\pi}{2}\\ \sin\frac{\pi}{2}& \cos\frac{\pi}{2}\end{bmatrix} which can be computed to \begin{bmatrix} 0 & -1\\ 1& 0\end{bmatrix}
If we apply it to a column vector \begin{bmatrix} 2 \\ 7\end{bmatrix}, we get \begin{bmatrix} 0 & -1\\ 1& 0\end{bmatrix}\begin{bmatrix} 2 \\ 7\end{bmatrix} = \begin{bmatrix} -7 \\ 2\end{bmatrix}, and if we plot both vectors on a 2D plane, they are, as expected, perpendicular to each other because the transformational matrix rotated the vector by 90^{\circ}.
I recommend you to try different rotations and even other examples listed in the wikipedia, then you will know what kind of transformation we can do with matrices, which is an idea of this lecture! Furthermore, you may use this tool for visualizing vectors as examplified in the tool’s 5-7th row (the first 4 rows shouldn’t be changed).
After trying more, I hope you will see that matrix can transform a 2D vector into another 2D vector, which is just what @paulinpaloalto meant by a \mathbb{R}^2 to \mathbb{R}^2.
If we look at your question again with the “rotational matrix”:
The matrix represents a rotational transformation. You might say it is a stack of two row or column vectors or a system of 2 equations, but I would not break it into two things because, for it to be rotational, it has to take that exact form, so I would rather consider them as a whole.
In fact, we can go more general to not just those wikipedia examples, but any matrix of any 4 numbers like yours:
It turns out that the above example can be decomposed into a (rotation with reflection), followed by two stretching, and then followed by another (rotation with reflection). The “decomposition” itself falls into an algebra topic called Singular Value Decomposition, or SVD. In principle, we can show how your example becomes the series of aforementioned transformations, but that is going to be a lot of details which I am not sure if it will be your focus, but if so, we can give that a try!
Cheers,
Raymond
Thanks @paulinpaloalto and @rmwkwok for your fast and detailed responses. I did some side reading and proceeded to the next lecture in Week 3 to get some better understanding.
The transformation matrix is the two vectors represented as columns. What was confusing to me was that professor chose to write ‘a’ and ‘b’ on top of those two columns in the matrix. If you look at the 2-D axis on the side of that slide, the axis ‘a’ and ‘b’ map to what we generally would refer as ‘x’ and ‘y’ axis. And so it almost makes one think does the ‘a’ column map to the coordinates on the ‘a’ axis?
I would think what the professor is going for is that ultimately the column ‘a’ in the matrix would be multiplied by the coordinate on the a-axis of the vector under transformation, and, similarly for ‘b’ column. This is what is represented by writing ‘a’ and ‘b’ on top of the matrix.
edit: some grammar improvements
All matrices and vectors are expressed using the standard Euclidean coordinates, so the first column of the transformation matrix are the coefficients that control what happens to the input vectors x coordinate and similarly for the second column expressing what happens to the y coordinate of each input vector.
Or to state the same point with more sophisticated terminology, everything is expressed using the standard basis vectors (1, 0) and (0, 1).
Also notice that if you look at the output of the transformation, then the x component of the output is affected by both coordinates of the input. And similarly for the y component of the output.