Matrix multiplication as composition | Chapter 4, Essence of linear algebra

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Strong rec Nεo Pλατo · 2020-02 · Mathematics Stack Exchange · on “Help with transformation matrices involving multiple transformations”

Have you heard of 3blue1brown? That YouTube channel is the only reason I understood linear algebra and more. Basically all you do is follow where the basis vectors go. Very smoothly and patiently, you will get an answer. [LINK] [this video] But if you want to confirm your math homework answers very quickly you could try this Desmos feature I dug up. [LINK] I won't compute the actual thing here cause Latex is stressful on a phone. But if this down here is what you worked out, then you'll be fine(matrix multiplication is inside out like function composition): $\begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{3} & 0\\ 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{\sqrt{2}}{2} & \fra

Recommends Fotis Rokanis · 2018-06 · Mathematics Stack Exchange · on “Visual representation of the $V/\ker f$ and first isomorphism theorem”

I'm trying to gasp the visual representation of $V/\ker f$ and in continuation the first isomorphism theorem in linear algebra. My understanding is that, given the representation of a vector space $V,$ $V/\ker f$ is created by "erasing" all the dimensions $f$ maps to $0.$ That's where I get stuck. Any help would be appreciated P.s The way I have linear algebra in mind is how it is presented in 3blue1brown's channel, if this is of any importance [this video]

Recommends user328442 · 2018-12 · Mathematics Stack Exchange · on “How is it that matrix multiplication represents linear transforms?”

Matrices themselves represent linear transformations and essentially tell one how a coordinate system changes. Matrix multiplication represents a composition of linear transformations (they are functions after all) and hence the product of two matrixes represents two consecutive coordinate changes done at once. Here is a link to neat video that might help with a geometric viewpoint to matrix multiplication: [this video]

Recommends Adel · 2019-02 · Mathematics Stack Exchange · on “3blue1brown's visually doing linear transformations composition”

I am trying to understand the step-by-step of visually calculating linear transformations following 3blue1brown chapter 4 (see youtube 3blue1brown ch 4 [this video] . Background: 3blue1brown describes how we can visually capture one transformation by simply recording where the base vectors ( $\hat{i}, \hat{j}$ ) land in the original coordinate system. So a rotation will produce new base vecors: $\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right]$ where the first column captures the new base for $\hat{i}$ and the second column is the new base for $\hat{j}$ . Following one transformation visually like this is straightforward. Conducting two transformations should produce similar results

Recommends Brian61354270 · 2019-03 · Mathematics Stack Exchange · on “Linear maps and matrix representations”

No, any basis will do. These [LINK] videos [this video] by Grant Sanderson may help you develop an intuition for why our choice in basis does not matter when describing a linear transformation. To summarize, each element $v_i$ of a vector $\mathbf v$ in $\mathbb R^n$ essential describes "how many" of the $i$ th basis vector to add up when constructing the vector $\mathbf v$ . In $\mathbb R^3$ , $$ \begin{bmatrix} a \\ b \\ c\end{bmatrix} = ae_1 + be_2 + ce_3 $$ where $\{e_1,e_2,e_3\}$ is any basis of $\mathbb R^3$ . Each column of a matrix $A \in \mathbb R^{n\times m}$ describes where the basis vectors of $\mathbb R^3$ "land" when moved through the transformation described by the matrix $A$ . The

Recommends vik1245 · 2019-10 · Mathematics Stack Exchange · on “Interpretation of multiplyng matrices with different dimensions geometrically”

I wanted to ask a question about the geometric interpretation of multiplying matrices but for non-identical dimensions. I'm currently studying an introductory course in matrices for my Quantum Physics courses in Chemistry and am using 3Blue1Brown's course linked here [LINK] . In chapter 3 [LINK] of his videos, he mentions multiplying matrices with vectors and showing this from a geometric perspective. In chapter 4 [this video] he shows how to do this with multiplying matrices together. Below is his example: $$\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & -2 \\ 1 & 0 \end{bmatrix}$$ This part makes sense. From reading right to left, considering vectors $\underline{i}$ and $\unde

Recommends Toby Mak · 2020-07 · Mathematics Stack Exchange · on “The reversibility of matrix multiplication problem in a proof by indution question about matrix raised to the nth power”

It might help to think of multiplying matrices as a series of function compositions. Then multiplying matrices is associative because the order of the compositions is the same. For more on what I mean, check out this 3B1B video [this video] .

Recommends Joz · 2020-09 · Mathematics Stack Exchange · on “Why multiply the rows of the first matrix by the columns of the second matrix?”

This video is very good at explaining the concept [this video] It might be useful to watch the whole series if you're not so sure on the intuition behind linear algebra

Recommends Shubham Gupta - TCH · 2021-04 · Mathematics Stack Exchange · on “How to figure out the transformation matrix for rotation and then sheer?”

I was watching this [this video] video. (Actually, I watched it 3 times because I couldn't understand it.) And right then, he showed that the trasformation matrix for rotation and then sheer is [LINK] I understood how he got the Rotation Matrix, that was the easy part, but how would you describe the Sheer matrix? Like, are the new coordinates dependent on the newly formed cartisian plane after Rotaiton or is it based on the initial cartian system. I tries both of them, but none seems to be the exact answer for final transformation. Infact, I get the transpose of the matrix instead. Can you explain how he got the Sheer Matrix.

Recommends Sophile · 2021-10 · Stack Overflow · on “Linear transformation: vectors landing at unintended places, manim”

@j1-lee The A=[[2 ,-1],[-1 , 3]] is to move i hat to (2,-1) and j hat to (-1,3). So, In a way, I am moving the entire vector space and not just a single vector. So, every vector in this vector space will be transformed. Have you watched this video [this video] ? It visualizes all these transformations.