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)

370 questions
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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 ...
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Proving that a linear isometry on $\mathbb{R}^{n}$ is an orthogonal matrix

I wish to prove that if $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is defined by $T(v)=Av$ (where $A\in M_{n}(\mathbb{R})$) is an isometry then $A$ is an orthogonal matrix. I am familiar with many ...
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Two idempotent matrices are similar iff they have the same rank

Definitions: For $F$ a field, $A,B\in F^{m\times n}$ are equivalent means there exist invertible matrices $Q\in F^{m\times m}$ and $P\in F^{n\times n}$ so that $B=QAP$, $A,B\in F^{n\times n}$ are ...
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The eigenvectors of the transpose operator

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know that the corresponding eigenvalues are $+1$ and $-1$, but I'm not sure how to find the eigenvectors of this ...
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Find matrix of linear transformation relative to new bases

If $T:\Bbb R^3\to \Bbb R^2$ is a linear transformation, and the matrix of $T$ = $\left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & -1\end{array}\right)$. If you use the basis $\{i,j,k\}$ ...
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Linear functional f(v)=0 imply v=0

Good day, I want to prove the following theorem: 1) For any nonzero vector $v \in V$, there exists a linear funtional $f \in V^*$ for wich $f(v) \neq 0$ I know that if $f$ is a lineal functional ...
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Reflect point across line with matrix

What is the transformation matrix that I multiply a point by if I want to reflect that point across a line that goes through the origin in terms of the angle between the line and the x-axis? In other ...
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How to find the matrix representation of a linear tranformation

How does one find the matrix representation of a linear transformation $T:V\to W$ with respect to the basis $B$ for $V$ and $D$ for $W$?
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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 ...
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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,...
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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 ...
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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 ...
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What is the mechanism of Eigenvector? [closed]

I have studied EigenValues and EigenVectors but still what I can't see is that how EigenVectors become transformed or rotated vectors.
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A linear transform of a closed set is closed

A linear transform of a closed set $E\subset \mathbb{R}^d \to \mathbb{R}^d$ is closed. I have seen a lot of similar questions here, but none of them exactly addresses the issue. Please if you find it ...
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If $T^m$ is diagonalizable for a $m\in\mathbb N$, then $T$ is diagonalizable.

Suppose that $V$ is a finite dimensional $\mathbb C$-vector space, and suppose that $T:V\rightarrow V$ is injective. If there is a $m\in\mathbb N$ such $T^m$ is diagonalizable, then $T$ is ...
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examples of linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
Dimension of $\mathbb{Q}$-vector space if a nonsingular linear transformation $T$ exists such that $T^{-1} = T^{2} + T$
We're given $V$ a finite dimensional vector space over $\mathbb{Q}$, $T$ a non-singular linear transformation of $V$ such that $T^{-1} = T^{2} + T$. The question has two parts. If I understand part (a)...