Questions tagged [linear-transformations]

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. (Def: http://en.m.wikipedia.org/wiki/Linear_map)

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, a linear function) is a mapping $V \to W$ between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. Reference: Wikipedia.

Linear maps can generally be represented as matrices, and simple examples include rotation and reflection linear transformations.

10279 questions
76
votes
6 answers

Equivalent Definitions of the Operator Norm

How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &=\sup\{ \lVert Av\rVert\;\colon\;…
57
votes
9 answers

If the field of a vector space weren't characteristic zero, then what would change in the theory?

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the this property, I am wondering what would we lose…
44
votes
6 answers

Reflection across a line?

The linear transformation matrix for a reflection across the line $y = mx$ is: $$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix} $$ My professor gave us the formula above with no explanation why it works. I am completely new to…
dsd
  • 465
  • 2
  • 6
  • 5
43
votes
3 answers

When can two linear operators on a finite-dimensional space be simultaneously Jordanized?

IN a comment to Qiaochu's answer here it is mentioned that two commuting matrices can be simultaneously Jordanized (sorry that this sounds less appealing then "diagonalized" :P ), i.e. can be brought to a Jordan normal form by the same similarity…
Mark
  • 5,488
  • 28
  • 30
39
votes
6 answers

The logarithm is non-linear! Or isn't it?

The logarithm is non-linear Almost unexceptionally, I hear people say that the logarithm was a non-linear function. If asked to prove this, they often do something like this: We have $$ \ln(x + y) \neq \ln(x) + \ln(y) \quad\text{and}\quad…
38
votes
6 answers

Prove that if V is finite dimensional then V is even dimensional?

Let $f:V \to V$ be a linear map such that $(f\circ f)(v) = -v$. Prove that if $V$ is a finite dimensional vector space over $\mathbb R$, $V$ is even dimensional. From what I can figure out for myself, if $V$ is finite dimensional, then every basis…
maths123
  • 521
  • 4
  • 7
34
votes
3 answers

Is a map that preserves lines and fixes the origin necessarily linear?

Let $V$ and $W$ be vector spaces over a field $\mathbb{F}$ with $\text{dim }V \ge 2$. A line is a set of the form $\{ \mathbf{u} + t\mathbf{v} : t \in \mathbb{F} \}$. A map $f: V \to W$ preserves lines if the image of every line in $V$ is a line in…
eepperly16
  • 6,347
  • 2
  • 21
  • 34
33
votes
4 answers

Prove that the axiom of choice is necessary in order to prove something else.

My mathematical background is perhaps a little lacking on this topic, but I've been searching and haven't come up with a satisfactory answer to this question. I have no idea how to approach the problem or if it has been answered. I have seen…
jodag
  • 745
  • 6
  • 16
33
votes
5 answers

If two matrices have the same eigenvalues and eigenvectors are they equal?

The question stems from a problem i stumbled upon while working with eigenvalues. Asking to explain why $A^{100}$ is close to $A^\infty$ $$A= \left[ \begin{array}{cc} .6 & .2 \\ .4 & .8 \end{array} \right] $$ $$A^\infty= \left[…
28
votes
4 answers

Why can't linear maps map to higher dimensions?

I've been trying to wrap my head around this for a while now. Apparently, a map is a linear map if it preserves scalar multiplication and addition. So let's say I have the mapping: $$f(x) = (x,x)$$ This is not a mapping to a lower or equal…
user3500869
  • 487
  • 4
  • 6
26
votes
15 answers

Is there more to explain why a hypothesis doesn't hold, rather than that it arrives at a contradiction?

Yesterday, I had the pleasure of teaching some maths to a high-school student. She wondered why the following doesn't work: $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$. I explained it as follows (slightly less formal) For your hypothesis to hold, it should hold…
Sanchises
  • 538
  • 4
  • 13
24
votes
3 answers

Is hyperbolic rotation really a rotation?

We define a $2\times 2$ Givens rotation matrix as: $${\bf G}(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) &\cos(\theta) \end{bmatrix}.$$ On the other hand, we define a $2\times 2$ hyperbolic rotation matrix as: $${\bf…
24
votes
10 answers

Is matrix transpose a linear transformation?

This was the question posed to me. Does there exist a matrix $A$ for which $AM$ = $M^T$ for every $M$. The answer to this is obviously no as I can vary the dimension of $M$. But now this lead me to think , if I take , lets say only $2\times2$ matrix…
avz2611
  • 3,548
  • 2
  • 16
  • 34
23
votes
0 answers

Lowest dimensional faithful representation of a finite group

How does one compute the lowest dimensional faithful representation of a finite group? This question originated in the context of given a finite group $G$: trying to find the lowest dimensional shape whose rotational/reflection symmetries form $G$.…
21
votes
2 answers

The range of $T^*$ is the orthogonal complement of $\ker(T)$

How can I prove that, if $V$ is a finite-dimensional vector space with inner product and $T$ a linear operator in $V$, then the range of $T^*$ is the orthogonal complement of the null space of $T$? I know what I must do (for a $v$ in the range of…
user62182
  • 310
  • 1
  • 2
  • 7
1
2 3
99 100