# 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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### Building topological equivalence between two vector fields

I'm reading about topological equivalence between different vector fields and I would like to know how to build these mappings. Let us consider one pedagogical example, suppose I have the following ...
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### How to get the state transition matrix by fundamental matrix when the fundamental matrix is not invertible? [closed]

I want to get the state transition matrix by the fundamental matrix solution $U(x)\in\mathbb{R}^{2\times2}$ as follow $\Phi(x,x_0)=U(x)U^{-1}(x_0)$. But since one column of $U(x_0)$ is $0$, the ...
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### How to use matrices to rotate two matrix to specific positions while keeping their relative position unchanged?

Suppose there are two vectors $\overrightarrow{a}, \overrightarrow{b}$ that define a plane. How do I find matrices that can be applied to both verctors, so that $\overrightarrow{a}$ will be rotated to ...
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### Is being isomorphic to dual hereditary?

Let $V$ be a vector space, and assume that $V$ is isomorphic to its dual, i.e., $V \simeq V^*$. Is every linear subspace $U$ of $V$ also isomorphic to its dual, i.e., $U \simeq U^*$? This is certainly ...
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### From Vandermonde matrix to Newton basis polynomial matrix (lower triangular)

Suppose I have an $n\times k$ Vandermonde matrix $V$ where the roots/points in $x \in \mathbb{R}^n$ are all distinct and matrix $V$ is of the form $$V= [1 \; x \; x^2 \ldots x^{k-1}]$$ where the ...
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### Does the direct sum of the image and kernel of the linear transformation form the original space? [duplicate]

Does the direct sum of the image and kernel of the linear transformation equal to the original space? I know if the linear transformation is a projection, the direct sum of the image and kernel of the ...
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### Linear maps from finite-dimensional spaces are continuous. Proof?

I am trying to find a non-circular path that leads me to the following result: Theorem 1. Linear maps from a finite-dimensional normed linear space (to possibly infinite dimensional spaces) are ...
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