Linear transformations and matrices | Chapter 3, Essence of linear algebra

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Strong rec Nεo Pλατo · 2020-02 · Mathematics Stack Exchange · on “Linear Algebra - finding angle between two vectors”

If you are learning linear algebra, here's a gem: [LINK] Best linear algebra intro ever! Watch everything. But anyway, you can associate that to a transformation from $\vec w_1 \to \vec w_2$ . [LINK] and [this video] . Those explain how it works. Thus rotation becomes a mapping of where $\hat \imath, \hat \jmath$ go when defined by a matrix. One column for $\hat \imath, \hat \jmath$ each. But in your case $\hat \imath=\vec w_1$ so I'll just approach it directly. Both basis vectors are on a unit circle. So any point on that circle is defined by the co-ordinates $(\cos(\theta), \sin(\theta)) \qquad \text{Check the diagram.}$ [LINK] So if $\hat \imath$ has been rotated, it has moved 120 degrees alon

Strong rec bubba · 2021-04 · Mathematics Stack Exchange · on “Looking for practical examples or books on linear transformations”

You’d probably have more luck with a computer graphics or CAD book (or web site) rather than a linear algebra book. Some good books are: Eberly [LINK] Goldman [LINK] Faux and Pratt [LINK] But they’re a bit expensive, and I expect you’ll find what you need on the internet. Just Google “transformation matrices”. For example, I think this guy [this video] is just brilliant.

Recommends Ben Grossmann · 2016-08 · Mathematics Stack Exchange · on “Alternate Characterization of Linearity”

This question is prompted by this video on matrices and linear transformations [this video] , which I highly recommend as a pedagogical tool. In it, the author characterizes linear transformations in the following way (I'm paraphrasing and formalizing) Define a line in a vector space to be a set of the form $L = \{u + t\,v: t \in \Bbb R\}$ for some vectors $u$ and $v$ (note: $L$ may consist of a single point) . That is, $S$ is an affine subspace of dimension at most $1$ . A function $T: \Bbb R^n \to \Bbb R^m$ is linear if: $T(0) = 0$ For any line $L \subset \Bbb R^n$ , the image $T(L)$ is a line in $\Bbb R^m$ I like this definition because of its geometric appeal and the fact that it manages to "

Recommends Bruno Reis · 2016-10 · Mathematics Stack Exchange · on “Linear transformation with linearly independent subsets of domain and codomain”

"Is there always a linear mapping $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ ?" No, and to see that you can apply the rank-nullity theorem: Let: $$ B_v = \left\{ v1,\cdots, v_n \right\}\\ B_w = \left\{ w1,\cdots, w_n \right\} $$ Where $B_v$ is basis for $\mathbb{R}^n$ and $B_w$ is a basis for $\mathbb{R}^m$. Assuming that $ker(T) = B_v$ and $im(T) = B_w$, we get: $$ \dim \mathbb{R}^n = \dim (B_v) + \dim (B_n)\\ n = n + m \Rightarrow \text{ False!} $$ With that you can conclude that the theorem says: No! But if you get a injective map, you'll se that the kernel of that function is $ \left\{ 0 \right\}$, and then, by the theorem it's going to be possible... So it's possible, but not always! To

Recommends Arthur · 2016-11 · Mathematics Stack Exchange · on “Find the matrix $A$ of transformation $T:\Bbb R^3\to \Bbb R^2$”

This video [this video] , while only talking about linear transformations $\Bbb R^2\to \Bbb R^2$, explains pretty well what such a matrix looks like.

Recommends Neil Aronson · 2016-12 · Mathematics Stack Exchange · on “Relationship between properties of linear transformations algebraically and visually”

I learned from 3Blue1Brown's Linear Algebra videos [this video] that a 2-D transformation is linear if it follows these rules: lines remain lines without getting curved the origin remains fixed in place grid lines remain parallel and evenly spaced I'm now going through linear algebra from a textbook, which lays out this definition of a linear transformation: T(u+v) = T(u) + T(v) T(cu) = cT(u) I'm wondering, is there a connection between these two ways of thinking of linear transformations? Do the visual ways of seeing 2-D linear transformations correspond to the formal definition when in 2-D?

Recommends Bunny · 2017-01 · Mathematics Stack Exchange · on “2 linear algebra questions”

This is not answer as requested . It is a explanation that can hopefully give you a more intuitive understanding to help you come up with your proofs. A linear function is any function that satisfies two properties: $ \gamma f(\vec x)= f(\gamma \vec x) $ and $ f(\vec x + \vec y)= f( \vec x)+ f(\vec y)$ where $\vec x \vec y $ are vectors and $\gamma $ is a scalar. You're instinct to think of a parabola is correct. What you want is a function that doesn't satisfy the two properties. Basically a linear function is a function that geometrically speaking can stretch and change directions of vectors but does not transform a straight line into a any graph with a curvature. This has to do with the f

Recommends sangstar · 2017-07 · Mathematics Stack Exchange · on “Question regarding the intuitive, visual interpretation of matrix multiplication”

I have a question on one of 3Blue1Brown's wonderful youtube videos regarding what matrix multiplication visually is. It can be found here. [this video] The part of the video I'm concerned with starts at roughly $6:20$. What I gathered, is that the narrator says that any matrix $A$ can be pictured as a transformation of basis vectors $\hat i + \hat j$ into $A$ by $IA$, where $\hat i + \hat j$ is the identity matrix and $A$ is what the basis vectors \begin{bmatrix} 1 & 0 \\ 0 & 1\\ \end{bmatrix}will now orient themselves as. When two matrices, $C$ and $D$ are multiplied together, I interpreted this as essentially taking $I$, multiplying it by $D$, then taking that transformed basis vector and multi

Recommends user266764 · 2017-12 · Mathematics Stack Exchange · on “How do I find the transformation matrix between two sets of points?”

@SpaceFace I could write a real tome about, so let me suggest a wondrous video which will clarify you everything: [this video]

Recommends Drewby · 2018-05 · Mathematics Stack Exchange · on “Why do you use the inverse matrix to find the image of a curve in the plane?”

Given a $2\times2$ linear transformation matrix and the equation of a curve in the plane (e.g. $x^2+y^2=1$), why does one use the inverse matrix to find the equation of the image of the equation after the transformation? I recently watched a video [this video] that explained how $2\times2$ matrices represent linear transformations in the plane. For example, if I wanted to find the image of the point $(1,0)$ after the transformation given by the matrix $$ \left[\begin{matrix}2&1\\-1&0\\\end{matrix}\right], $$ I would simply multiply: $$ \left[\begin{matrix}2&1\\-1&0\\\end{matrix}\right]\left[\begin{matrix}1\\0\\\end{matrix}\right]=\left[\begin{matrix}2\\-1\\\end{matrix}\right]\rightarrow(2,-1). $$