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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)

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49
votes
9answers
3k views

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 ...
39
votes
6answers
11k views

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 + ...
34
votes
6answers
3k views

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,...
30
votes
5answers
75k views

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 ...
30
votes
3answers
2k views

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 ...
28
votes
5answers
7k views

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 & ...
26
votes
15answers
3k views

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 ...
26
votes
4answers
5k views

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: $$f(x) = (x,x)$$ ...
18
votes
2answers
3k views

A real function which is additive but not homogenous

From the theory of linear mappings, we know linear maps over a vector space satisfy two properties: Additivity: $$f(v+w)=f(v)+f(w)$$ Homogeneity: $$f(\alpha v)=\alpha f(v)$$ which $\alpha\in \...
17
votes
4answers
132k views

Find the standard matrix for a linear transformation

If T: $\Bbb R$3→ $\Bbb R$3 is a linear transformation such that: $$ T \Bigg (\begin{bmatrix}-2 \\ 3 \\ -4 \\ \end{bmatrix} \Bigg) = \begin{bmatrix} 5\\ 3 \\ 14 \\ \end{bmatrix}$$ $$T \Bigg (\begin{...
17
votes
8answers
14k views

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 ...
17
votes
2answers
325 views

Orbits of vectors under the action of $\mathrm{GL}_n(\mathbb Q)$

Context. While working on a larger proof, I would love to have the following lemma, but I can't even decide if it's true or not. The question. We consider the action of $\mathrm {GL}_n(\mathbb Q)$ ...
16
votes
7answers
6k views

Why is a square root not a linear transformation?

The question says: Prove that the function $f(x)=\sqrt{x}$ is not a linear transformation (particularly $\sqrt{1+x^2}≠1+x$) I think that this is because the exponent of $\sqrt{x}$ is $1/2$, and ...
15
votes
3answers
1k views

What are some meaningful connections between the minimal polynomial and other concepts in linear algebra?

I’ve found that the most effective way for me to deeply grasp mathematical concepts is to connect them to as many other concepts as I can. Unfortunately, I’m seeing neither the importance nor the ...
15
votes
3answers
2k views

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 ...
14
votes
8answers
2k views

Why doesn't the definition of dependence require that one can expresses each vector in terms of the others?

I was reviewing my foundations on linear algebra and realized that I am confused about independence and dependence. I understand that by definition independence means: A set of vectors $\{x_1,\...
14
votes
6answers
303 views

Simple proof that if $A^n=I$ then $\mathrm{tr}(A^{-1})=\overline{\mathrm{tr}(A)}$

Let $A$ be a linear map from a finite dimensional complex vector space to itself. If $A$ has finite order then the trace of its inverse is the conjugate of its trace. I know two proofs of this fact, ...
14
votes
4answers
297 views

What is the dimension of $\{X\in M_{n,n}(F); AX=XA=0\}$?

Let $A$ be a fixed $n\times n$ matrix over a field $F$. We can look at the subspace $$W=\{X\in M_{n,n}(F); AX=XA=0\}$$ of the matrices which fulfill both $AX=0$ and $XA=0$. Looking a these equations ...
13
votes
2answers
679 views

Intuitive geometric explanation: existence of eigenvalue in odd dimension real vector space.

I'm looking for an intuitive geometric explanation for the fact that given an odd dimensional real vector space $W$ and an endomorphism $T:W \rightarrow W$, there exists a real eigenvalue of $T$. I'm ...
13
votes
2answers
210 views

Does a family of linearly independent injective maps have a vector with linearly independent images?

Let $V,W$ be finite dimensional vector spaces over an infinite field $k$. Fix a positive integer $n \leq \dim(W), \dim(V)$. Given $n$ injective linear maps $f_i:V\rightarrow W$, such that the $f_i$ ...
12
votes
3answers
9k views

What is the difference between linear transformation and linear operator?

What is the difference between linear transformation and linear operator? In our linear algebra class, we learned that, if $$\textbf{T}:\textbf{V}\rightarrow\textbf{W}\quad\vec{v},\vec{u}\in\textbf{V}...
12
votes
5answers
256 views

Ill-known/original/interesting investigations on/applications of inversion (the geometric transform)

Inversion transform with center (or pole) $C$ and power $k^2$ is defined by: $$\tag{1}J_{C,k}:M \leftrightarrow M' \ \ \ \ \ \iff \ \ \ \ \ \ \ \vec{CM'}=\frac{k^2}{||\vec{CM}||^2} \ \vec{CM} $$ It ...
12
votes
2answers
2k views

Intuition behind the definition of linear transformation

I have studied that given vector spaces $V_1$ and $V_2$, a function $T:V_1 \rightarrow V_2$ is called a linear transformation of $V_1$ into $V_2$, if following two properties are true for all $u, v \...
12
votes
5answers
618 views

What is an intuitive way to understand the dot product in the context of matrix multiplication?

I was trying to understand where it came from that each row in a matrix multiplication is a dot product, as in: $$ Ax = \left( \begin{array}{ccc} a_{1}^T \\ \vdots \\ a_m^T \end{array} \right)x = \...
12
votes
0answers
658 views

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 ...
11
votes
2answers
1k views

Show that an open linear map between normed spaces is surjective.

Let $X,Y$ be normed spaces and $T:X\to Y$ is an open linear map. Show that $T$ is surjective. In order to show $T$ is surjective let's take $y_0\in Y$ and assume the contrary that $Tx\neq y_0\forall ...
11
votes
3answers
16k views

How do I find a dual basis given the following basis?

$V = \Bbb{R}^3$ and has basis $\mathcal{B} = \{\vec{e_1}-\vec{e_2},\vec{e_1}+\vec{e_2},\vec{e_3}\}$ How do I find the dual basis? This is not homework, but an example that I am struggling to grasp. ...
11
votes
0answers
190 views

Coordinate free proof that $\operatorname{trace}(A)=0\:\Longrightarrow\:A=BC-CB$

As you probably know, the trace function on square matrices has the property that $$\operatorname{trace}(AB-BA)=0\,.$$ You might also know that the converse is true: $$\operatorname{trace}(A)=0\;\text{...
10
votes
6answers
2k views

Must an injective or surjective map in an infinite dimensional vector space be a bijection?

If we have some finite dimensional vector space V and linear transformation $F: V\to V$, and we know that F is injective, we immediately know that it is also bijective (same goes if we know that F is ...
10
votes
1answer
455 views

How to understand the exponential operator geometrically?

Consider the geometric interpretation of an orthogonal matrix, a projection matrix, a (Householder) reflector, or even just matrix-vector multiply in general. A matrix takes a vector from a vector ...
10
votes
1answer
312 views

Are inner product-preserving maps always linear?

Let $E,F$ be Pre-Hilbert spaces and $T: E \rightarrow F$ be a map that preserves the inner product, that is $\langle Tu , Tv \rangle = \langle u , v \rangle$ for all $u,v \in E$. Must it be true that $...
9
votes
2answers
474 views

This theorem about matrices of linear maps doesn't look correct.

Consider the following theorem: Theorem. Let $f\colon L\to M$ be a linear mapping of finite-dimensional vector spaces. Then there exist bases in $L$ and $M$ and a natural number $r$ such that the ...
9
votes
3answers
1k views

Should a linear function always fix the origin? [duplicate]

I became very confused about linear functions after reading this question What is the difference between linear and affine function In the comments it says that $F(x)=2*x+4$ is NOT a linear function ...
9
votes
5answers
847 views

Compute $\det{T}$ where $T(X)=AX+XA$

Consider the linear transformation $T:V\to V$ given by $T(X) = AX + XA$, where $$A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}.$$ Compute the determinant $\det T$. ...
9
votes
1answer
468 views

Given $T \in L(X,Y)$, show the equivalence between: existence of $S$ such that $S(T(x))=x$, and $T$ being injective with $T(X)$ complemented in $Y$

Given $X,Y$ Banach spaces and $T \in L(X,Y)$, show that the following sentences are equivalent: A) there exists $S \in L(Y,X)$ such that $S(T(x))=x$ for all $x \in X$. B) $T$ is injective and $T(X)$ ...
9
votes
2answers
255 views

Is there an injective operator with a dense nonclosed one-codimensional range?

Let $H$ be an infinite dimensional separable Hilbert Space. Is there an operator $A\in B\left( H\right) $ such that $Im\left( A\right) \neq \overline{Im\left( A\right) }=H$, $ codim\left(Im% \left( A\...
9
votes
4answers
840 views

Under what conditions is a linear automorphism an isometry of some inner product?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and that $T: V \to V$ is a (linear) isomorphism. When is it possible to construct an inner product on $V$ making $T$ an ...
9
votes
2answers
4k views

Is the number of linearly independent rows equal to the number of linearly independent columns?

For any matrix the column rank and row rank are equal. As I understand it rank means the number of linearly independent vectors, where vectors is either the rows or columns of the matrix. This seems ...
9
votes
2answers
295 views

Is a function which preserves zero and affine lines necessarily linear?

My question is directly inspired by this other recent question, but I was trying to figure out whether or not it holds for $\mathbb R$. This led me to two questions. Let $n \ge 2$ be an integer (we're ...
9
votes
1answer
176 views

Bases for symmetric polynomials

I’ve been playing with symmetric polynomials, as one does, and I’ve run into something that must be familiar, but I can’t find anything about it. To present the idea, I’ll work with symmetric ...
9
votes
1answer
275 views

How to find a symmetric matrix that transforms one ellipsoid to another?

Given two origin-centered ellipsoids $E_0$ and $E_1$ in $\mathbb{R}^n$, I'd like to find an SPD (symmetric positive definite) transformation matrix $M$ that transforms $E_0$ into $E_1$. Let's say $...
9
votes
0answers
126 views

Finite group of “linear substitutions”

From what I can tell, a linear substitution is an operation on a set of variables $x_1,\ldots,x_n$ which sends them to a new set of variables $y_1,\ldots, y_n$ via a linear transformation $$\vec{y} = ...
9
votes
0answers
247 views

Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
9
votes
1answer
107 views

Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...
8
votes
2answers
839 views

Translating an Italian exercise precisely

I am trying to help a friend with his algebra course. However, his exercises are in Italian, and unfortunately he translates them poorly for me since he does not know the mathematical terms in ...
8
votes
2answers
39k views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: $$...
8
votes
2answers
2k views

How can I get eigenvalues of infinite dimensional linear operator?

What I want to prove is that for infinite dimensional vector space, $0$ is the only eigenvalue doesn't imply $T$ is nilpotent. But I am not sure how to find eigenvalues of infinite dimensional linear ...
8
votes
2answers
338 views

If a linear map $T$ has a $k$-dimensional invariant subspace, does it admit an $n-k$ invariant subspace?

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n-1$ be fixed. Let $T: V\to V$ be a linear map, and suppose that there exists a $k$-dimensional $T$ invariant subspace of $V$. ...
8
votes
3answers
19k views

Finding kernel and range of a linear transformation

We are given: Find $\ker(T)$, and $\textrm{rng}(T)$, where $T$ is the linear transformation given by $$T:\mathbb{R^3} \rightarrow \mathbb{R^3}$$ with standard matrix $$ A = \left[\...
8
votes
2answers
478 views

Let $A$ be a symmetric matrix of order $n$ and $A^2=0$ . Is it necessarily true that $A=0$

Let $A$ be a symmetric matrix of order $n$ and $A^2=0$ . Is it necessarily true that $A=0$ . My approach : I tried to experiment with some $2\times 2$ matrices but never gotten any far . Now, ...