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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://...

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Let v in Im(p). Compute p(v).

Let $B = (1, X, X^2)$ be a basis for $R_2[X]$ and $p ∈ L(R^2[X])$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)$ and $p(X^2) = \frac{1}{3} (−1 − X ...
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Compute $rk(p)$ using Gauss reduction on $A$. Compute $dim(ker(p))$.

Let $B = (1, X, X^2)$ be a basis for $R_2[X]$ and $p ∈ L(R^2[X])$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)$ and $p(X^2) = \frac{1}{3} (−1 − X ...
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17 views

Linear transformations - 2 opposite claims, solution attempt included

Suppose $V, W$ are vector spaces and $ T: V \to W $ is a linear transformation. $v_1, v_2, ... , v_k \in V$. Prove or disprove: If $span( v_1, v_2, ... , v_k) = V$, then $span(T(v_1), ... , T(v_k)...
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27 views

Prove that p ◦ p = p. (Representing a Linear Transformation as a Matrix)

Let $B = (1, X, X^2)$ be a basis for $R_2[X]$ and $p ∈ L(R^2[X])$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)$ and $p(X^2) = \frac{1}{3} (−1 − X ...
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2answers
30 views

Linear map $L \neq O$ having trivial image only at $L^2=L \circ L$

From S.L Linear Algebra: Let $L:ℝ^2 \rightarrow ℝ^2$ be a linear map such that $L \neq O$ but $L^2=L \circ L=O$. Show that there exists a basis $\{A, B\}$ of $ℝ^2$ such that, $L(A)=B$ and $L(B)...
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Prove that in QR factorization, R is the change of coordinate matrix from the standard basis to the basis of the vectors in Q

I really cannot understand the proofs given to me by many different people, and I wasn't able to find a proof of this online. Let A be a matrix consisting of n linearly independent vectors, forming a ...
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29 views

Insightful proof for $UV^T = \sum_{i=1}^n u_iv_i^T $

Let $U = [u_1, u_2,\dots ,u_n] \in \mathbb{R}^{m\times n}$ with $u_i \in \mathbb{R}^m$ and $V = [v_1, v_2,\dots ,v_n] \in \mathbb{R}^{p\times n}$ with $v_i \in \mathbb{R}^p$. Then $$UV^T = \sum_{i=1}^...
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Finding the matrix form of a linear operator with a non canonical vector space $V$?

Given a vector space $V =\{a\cos(t) + b\sin(t) + ct\sin(t)\}$ with $a,b,c$ all real, a basis $B = \{cos(t),sin(t),t\sin(t)\}$ and a Linear operator $Lf(t) = f''(t)$, how would I write ${\lbrack L\...
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28 views

Linear operator and inner product

Theorem: Let $V$ be an inner product finite space with an orthonormal basis $\mathcal B$. Let $L$ be an operator on $V$, and let $A = [L]_\mathcal{B}$, the matrix associate to $L$. Then the matrix ...
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48 views

How many solutions $Ax=0$ has

Problem How many solutions equation $Ax=0$ has when $A$ is defined as: $$ A = \begin{bmatrix} 2 & 3 & 5 \\ 1 & 4 & 0 \\ -4 & 2 & 1 \end{bmatrix} $$ Attemtp to solve ...
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Find an orthonormal basis for the linear subspace of $\mathbb{R}^4$ [on hold]

I need to find an orthonormal basis for the linear subspace of $\mathbb{R}^4$ generated by: $$ \begin{pmatrix}1\\1\\0\\0\end{pmatrix} \;,\; \begin{pmatrix}2\\1\\1\\2\end{pmatrix} \;,\; \begin{pmatrix}...
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How to show a polynomial transformation defines a linear transformation [on hold]

The question is: Show that the map $T:R[x]_3\to R[x]_3$ given by $$T(p(x))=p''(x)-xp'(x)+2p(x)$$ defines a linear transformation. I know I need to show it's closed under addition and scalar ...
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Help Finding the matrix for T relative to standard bases [on hold]

Define T: P2--->R3 by the equation T(p(t))= [p(-1), p(0), p(1)] Find the matrix for T relative to the standard bases of P2 and R3 Sorry in advanced for the formatting. I don't know how to do the ...
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1answer
26 views

How to find CDF of $Y=|X|\wedge 2$ with $X\sim Laplace(\lambda)$

Given $X$ a bilateral exponential with density $f_X(x)=\frac{1}{2}e^{-|x|}, \forall x\in \mathbb{R}$ and $\lambda=1$, i have to find CDF of $Y=|X|\wedge 2$. I know that $Y$ is not a monotonic ...
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1answer
25 views

Find a matrix which has the kernel same as a set

The question is Let $$W = \{(x_1, x_2 , x_3 , x_4)^T : x_1 + x_2 + x_3 + x_4 = x_1 + 2x_2 + 3x_3 + 4x_4 = 0\}$$ Find a matrix $A$ such that $W$ = $\ker A$. And the answer is $$\pmatrix{1 & 1 ...
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Dimension of the range space [on hold]

Let $T : V → V$ be a linear map on a finite dimensional space of dimension $n$ such that the norm of $T(v)$ is bounded by the norm of $v.$ Find the dimension of the range of $T − √2 I$. I don’t have ...
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2answers
45 views

Orthogonal transformation with additional constraints

Let $A$ be an orthogonal matrix, i.e. $AA^{T}=\mathbb{I}$. It is given that $A$ satisfies an additional constraint, $AMA^{T}=PMP^{T}$, where $P$ is some permutation matrix and $M_{ij}=sgn(i-j)$. Can $...
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If T :Rn →Rm is a linear transformation and m=n then T is an isomorphism [on hold]

Is it true? I know that it should be invertible but ihow do i know inthis case
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23 views

Eigenspace and dimensions of linear transformation in complex plane

Suppose V is a finite-dimensional complex vector space and T : V → V is a linear transformation which satisfies $T^{2} = T$. Prove that for any vector v $\in$ V , the vector T(v) is contained in the ...
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1answer
19 views

Rank of Linear Transformation Preserved

Show that the linear transformation with rank $m$ on $n$-dimensional subspace $V$ can be expressed as the sum of $m$ linear transformations with rank $1$. Since any linear transformation can be ...
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1answer
17 views

Solution for Linear Transformation question seems wrong.

I'm studying with a Linear Algebra book which presents the following question: Be $A:E\rightarrow F$ a linear transformation. If the vectors $Av_1,\ldots,Av_n \in F$ are LI, then prove that $v_1,\...
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1answer
31 views

If rank$(T)\le$rank$(T^3)$, then intersection of range and null space of $T$ is zero

If $T:V\to V$ be linear trannsformation on a vector space $V$ with rank$(T)\le$ rank$(T^3)$. Then, what can be said about the null space and range of $T$. Typically, do they have empty intersection, ...
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1answer
21 views

Regarding Hahn Banach theorem and supporting hyperplane

In the above image from the book Linear Analysis by Bela Bollobas, corollary 7 gives the first consequence to the Hahn Banach Theorem. In the paragraph below corollary 8 they define a supporting ...
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2answers
33 views

What properties of a linear map can be determined from its matrix?

I am currently taking a proofs based linear algebra course for math undergraduates. It's been almost two years since I took a more computational linear algebra course (solving matrix equations, ...
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32 views

Homogeneity does not suffice for a map between vector spaces to be linear

The following problem is taken from Sheldon Axler's book Linear Algebra Done Right, more precisely Exercise 1. from Chapter 3: Problem: Give an example of a function $f : \mathbb{R}^2 \to \mathbb{R}$ ...
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1answer
13 views

Mappings to spaces with different numbers of dimensions

The first diagram first question second question I seem to have a good understanding of linear transformation and linear algebra yet I fail to grasp this completely. Question 1) Please explain ...
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46 views

Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
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1answer
38 views

Bounds for the rank of the sum of two linear maps

The following is Exercise 3.15 from the German textbook Lineare Algebra by Hans-Joachim Kowalsky and Gerhard O. Michler: Let $\varphi$ and $\psi$ two linear maps from a finite-dimensional vector ...
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1answer
40 views

Questions on symmetric matrices and skew-symmetric matrices

Let $A$ be a $3\times 3\;$ symmetric matrix. Let $U$ be the set of all $3\times 3\;$ skew-symmetric matrices. Let $T : U\to U$ be defined as $T(B)=AB+BA.$ Prove that $T$ is bijective iff the sum of ...
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1answer
22 views

How to apply this linear transformation $(x:=2x+1)$ on R-sequence result to generate points in range $[-1,1]$?

I want to used R-sequence proposed by Martin Roberts to generate points in range [-1,1]. In this post, Martin Roberts mentioned that: ... to convert to a range of [-1,1], simply apply the linear ...
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1answer
33 views

Properties of Positive Real Functions

I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is ...
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2:1 Diametric rotation matrix for a 2D orthographic projection

I asked this question in the game dev stack exchange, but didn't get a response. This is more of a math question with an application in game development so I hope it's alright if I ask here. I'm ...
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1answer
21 views

Proof of relation between Normal and Chi-square

Let $X\sim N(0,1)$, i want to determine the distribution of $Y=X^2$. By definition, the density of $X$ is $f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2},\forall x\in \mathbb{R}$...
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3answers
26 views

Finding the kernel of the following linear transformation [closed]

Let $A=(\frac{2x}{a^2},\frac{2y}{b^2},\frac{2z}{c^2})$ be a linear transformation $\mathbb{R}^3 \rightarrow \mathbb{R}^3$(where $a,b,c$ are constants $\in \mathbb{R}^3$). Find $ker(A)$. How do I find ...
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18 views

Isomorphism and dimension exercise clarification.

I'd need a clarification on one of the statements here below. Given that V is the set of linear operators f: $\mathbb{R}^k \rightarrow \mathbb{R}^n$ , and I'm trying to prove that V has same ...
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0answers
54 views

Operator in finite dimension spaces: $T: X \rightarrow Y$ is open iff $T(X)=Y$. [closed]

To the functional analysis course I am attending I am given to solve this exercise. Given $X$ and $Y$ normed spaces, $Y$ with finite dimension. Show that a linear operator $T: X \rightarrow Y$ is an ...
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33 views

How to prove non-linear functions use copying/deleting? [on hold]

Consider a ring $\pmb{R} : (R, +, \bullet)$ where $r$ is the underlying set and $+$ and $\bullet$ are the two binary operations of addition and multiplication satisfying the usual ring axioms. Now, ...
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1answer
35 views

Proving existence of a linear transformation with given properties

The question is as follows: Suppose that $V$ is a vector space over $\mathbb{C}$ of dimension $3$. Fix a non-zero vector $v\in V$ and define $$U:=\{T\in\mathcal{L}(V):v\mbox{ is and eigenvector ...
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1answer
17 views

How do I determine whether a function/map T is one-to-one and/or onto?

Given $T(x,y,z) = (xy, yz, xz)$, determine whether $T$ is one to one and/or onto. So far, I have come up with a contradiction that proves $T$ is not one to one: $T(-1, 1, -1) = (-1, -1, 1)$ and, $T(...
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1answer
25 views

Two labelled sets of vectors in $\mathbb{R^n}$ with same pairwise dots products

Given $\{x_i\}_{1 \leq i \leq m}$ and $\{y_i\}_{1 \leq i \leq m}$ such that $x_i \cdot x_j$ = $y_i \cdot y_j, \forall 1 \leq i,j \leq m$, what can I conclude about the two sets of vectors? Clearly ...
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1answer
30 views

Transform a point with a $4\times4$ matrix

So I have a series of points and a $4\times4$ transformation matrix. I have a bit of understanding of whats going on but what exactly am I multiplying and in what order? I get the top left $3\times3$ ...
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1answer
18 views

Can a linear isometry always be expressed in terms of an orthogonal matrix?

Is the following true? Let $S: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation such that $||S(v)|| = ||v|| \ \text{for all} \ v \in \mathbb{R}^n$, where $||\cdot||$ denotes the Euclidean ...
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4answers
47 views

If $A^2=I$ and $\lambda \not =-1$, then $A=I$ [duplicate]

Given an $n\times n$ matrix $A$ with $A^2=I$, assume that $-1$ is not an eigenvalue of $A$. Prove $A=I$. Proof attempt: Since $A^2=I,$ we have $A^{-1}=A$. Using that fact that $\det(A)=\prod_{i=1}^n\...
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1answer
23 views

Find the matrix of $T$ with respect to standard basis of $V$.

Let $T:V\to V$ be the rotation by an angle $\theta$ counterclockwise in the plane passing through the origin perpendicular to $(1,2,3)$ where $V=\Bbb R^3$ Find the matrix of $T$ with respect to ...
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1answer
46 views

why preserving norm is equivalent to preserving inner product in rigid body transformation

Define rigid body motion as following transformation $g$ $$g:\mathbb R^3 \to \mathbb R^3$$ such that $$|g(v)|=|v|,\forall v \in \mathbb R^3$$ $$g(u) \times g(v) = g(u\times v)$$ according to ...
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1answer
27 views

Proving a Linear Transformation if onto/one-to-one for vector spaces

Given two vector spaces $V$ and $W$ of finite dimensions and a linear transformation $T: V \rightarrow W$, let $m = dim(V)$ and $n = dim(W)$. Show that if $T$ is one to one, then $m \leq n$ and if $T$ ...
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1answer
30 views

Find a linear transformation defined by $T(0,1,2) = (3,1,2)$ and $T(1,1,1) = (2,2,2)$. [on hold]

The transformation here (as per my calculations) will be $T\colon U \rightarrow V$ such that $T(x,y,z) = (y+z, 3y-z, 2y)$ where $z=-x+2y$. Now what should $U$ be $\Bbb R^3$ or a subset of $\Bbb R^3$?...
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1answer
21 views

What do “projections” out of tensor products look like?

Convention. If $R$ is a relation $X \rightarrow Y$, what I mean is that $R$ is a subsets of $X \times Y$. We say that $X$ is the domain of $R$ and that $Y$ is the codomain. We'll write $x \overset{R}{\...
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1answer
16 views

Definition of scalar product in relation to projections

So if $x \in X$ is an element in a vector space $X$, then $\forall x \in X$: $$x = e_ix^i$$ where $e_i$ is a basis for $X$. However, I encountered that this is equivalent to: $$x = e_i (e_i, x)$$ ...
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1answer
24 views

Measuring “concentration” in an expansion

Suppose we have a vector $v \in \mathbb{R}^{n}$ which we expand in some orthonormal basis $\{g_{m}\}$ of $\mathbb{R}^{n}$: \begin{align*} v = \sum_{m=1}^{n} a_{m}g_{m} \end{align*} I want to measure ...