# 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)

6,274 questions
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### How to understand root subspace in linear algebra

According to the definition of the root subspace,there exists an exponent k that$(\mathcal A-\lambda\epsilon)^k v=0$,but how to confirm this exponent?If it is an n*n matrix,what happens when the ...
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### How to define pivot columns?

When you use Gaussian elimination to solve a homogeneous system of linear equations, you end up with "pivot variables" and "non-pivot variables". The non-pivot variables have the property that they ...
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### Proving a matrix inequality given another inequality

Suppose that for the $2$-norm, we have $||A||_{2} < 1$. Show that $||I - A^{T}A||_{2} < 1.$ Assume $A$ is invertible. I don't know how to solve this problem. I'm studying for an exam. I know ...
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### Composition of rotation and reflection

Suppose you find a spatial isometry and you want to classify it. You find out the only real eigenvalue is -1. Also, there is a unique fixed point c. In this type of cases i was told that we have a ...
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### How to get a new matrix with the diagonal elements of another one, and zeros in the rest of the entries? [closed]

I'm NOT interested in how I can do this in MATLAB. I would like to know how, given a matrix $A\in\mathbb{R}^{n\times n}$, I can extract only the elements on its diagonal, put them in a new matrix $B$ (...
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### If $\Sigma^{-1}=(A^{-1})^TA^{-1}$, then why does $|A^{-1}|=|\Sigma|^{-1/2}$?

In the derivation of the joint pdf of $f_\textbf{X}(\pmb{x})$, where $\textbf{X}=\pmb\mu+A\pmb Z$ and $\textbf{X}\sim~N_n(\pmb\mu,\Sigma)$, there is a step I do not understand. In particular, it is ...
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### Meaning of the determinant of the restriction of a linear map

Suppose $T : \mathbb{R}^n \to \mathbb{R}^n$ is a linear map and let $U \subset \mathbb{R}^n$ be a $d$-dimensional subspace where $0 < d < n$ and $\ker T = U$. I was wondering how to make sense ...