Three-dimensional linear transformations | Chapter 5, Essence of linear algebra

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Recommends Ben Grossmann · 2017-07 · Mathematics Stack Exchange · on “Visualizing Linear Maps”

You might find this video to be helpful [LINK] , and also this one [this video] from the same author on the 3-dimensional case.

Recommends Physor · 2020-09 · Mathematics Stack Exchange · on “Geometrical interpretation of complex differentiability”

I try to explain the meaning of $f'(z)dz$ for complex differentiable functions at a point . Let the complex numbers be interpreted in the plane as arrows starting from the origin and making some angle with $x$ -axis. Their addition is clear (I hope) but their multiplication has an arrow with a lenght as the product of the lenghts and an angle as the sum of the angles. Let us write $dz = re^{i\varphi}$ and $f'(z) = Re^{i\Phi}$ . Their product is $$ f'(z)dz = re^{i\varphi} Re^{i\Phi} = rR e^{i(\varphi + \Phi)} $$ This product should represent the small change in $f(z)$ as response to a sufficiently small change $dz$ in $z$ . That is $$ \Delta f(z) \approx df(z) = rR e^{i(\varphi + \Phi)} $$ No

Cites tomi · 2021-01 · Mathematics Stack Exchange · on “Linear vector transformations in 3Blue1Brown with basis vectors”

I am looking at the 3Blue1Brown vector transformation video [this video] and at 2:17 [this video] he gives an example of a transformation that rotates space 90 degrees over the y-axis. Based on the intuition I have it seems weird because he says that $\hat{i}, \hat{j}$ and $\hat{k}$ are all basis vectors. Therefore in a 3-dimensional Euclidean space the vectors should have the values $(1,0,0), (0,1,0)$ and $(0,0,1)$ respectively. Now I have the understanding that every transformation can be expressed as some scalar multiplicaton of the basis vectors. For exaple if I have a vector $\overrightarrow{v} = (2,3)$ where the $2$ and $3$ are the coordinates of the vectors that can be represented in relation t