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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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### 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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### 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 ...
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### 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: $$f(x) = (x,x)$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...