Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Diagonalisable linear operator on infinite-dimensional vector space: definition problem

How to define a linear operator on an infinite-dimensional vector space to be diagonalisable? I have tried to search for the definition on internet but it seems not very fruitful as all are about ...
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series problem feedback needed

this is a example from my lecture notes and while studying i cannot understand how the final answer is found. the particular place where i am confused i have circled. shouldn't it be -1/n(n+1) ?
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Magnitude of the value of a line at a point which does not lie on it

This may be trivial but I had never thought of it before. I know that the sign of the value represents on which side of a line a point lies but don't know what the magnitude of that value represents. ...
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Prove that minimal polynomial of $T =$ minimal polynomial of$ A$, if$ T(B) = AB$

Let $A \in M_{2×2}(F) $and define $T(B) = AB$ for $B \in M_{2×2}(F)$. (i)Prove that $cT(x) = (cA(x))^2$. (ii) Prove that $mT(x) = mA(x)$. (iii) Prove that $T$ is diagonalizable if and only if $A$ ...
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Matrices of rank 1

Let the $n \times n$ complex matrix $A$ have rank 1. Prove: $A^2 = c\cdot A$ for some scalar $c$. What I know is that all matrices having rank 1 have rows based on a scalar multiple of the other. ...
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Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v

I wrote a proof for the above question, but I am not sure whether it is right or not since I assumed linear independence. Here's the proof: Let $u$,$v$ be linearly independent vectors in $V$. ...
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1answer
34 views

A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding $A^{-1}$

The question is: A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding A^-1. I have looked at other similar questions on this site: 1. Here 2. and Here But they use ...
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Prove that the matrix generated from a bounded non-zero set and prime numbers has a non-zero determinate

Given a finite set $S \subset \mathbb{Z}$ with the following conditions: $$\forall s \in S, s \neq 0 $$ $$ \sum_{s \in S} |s| < \prod_{n=1}^{|S|} P_n $$ Let A be the $|S|x|S|$ matrix with entry ...
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How can I backward substitute this 4x4 guassian elimination

I was trying to solve this guassian elimination and I think i have it into the required 'staircase' pattern needed for guassian elimination. And here it is after forward elimination: 1 2 0.5 | 3.5 ...
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Coding a Message using Matrices

Ok, so this problem I've been working on for the past hour, with no answer. In coding a message, a blank space was represented by 0, an $A$ by 1, a $B$ by 2, a $C$ by 3, and so on. The message was ...
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elementary matrix

$$\begin{bmatrix} -5 & 1 \\ 1 & 0\end{bmatrix} $$ Why is this not the elemtary matrix? Why do we not change the first row of the second row place and thus be elementary matrix?
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Minimize distance to a given point subject to a number of linear inequality

I'm trying to find a point that has minimal distance to a known point and satisfies a number of linear inequality. Example in two dimensions and one inequity: $min\{$distance to $(50,70)$ | ...
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23 views

Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that ...
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Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$.

Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$. I'm not sure how to start proving this. I don't ...
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Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation.

Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation. I'm sorry, but this question throws me off in many ...
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25 views

Linear map - how to show this?

Assuming that I have a map $A: \mathbb{R}^2 \rightarrow \mathbb{R}$ and we have $A(-x,x) = -A(x,x)$ and $A(x+y,x) = A(x,x)+ A(y,x)$. Is this sufficient to conclude that $A( \lambda x+y ,x ) = \lambda ...
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1answer
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Spans and Dot Product: Findin the linear combination

Suppose $(v_1, v_2, v_3)$ is a set of vectors mutually perpendicular. Assume that $\|v_1\|= \sqrt{27}\quad \|v_2\| = \sqrt{14}\quad \|v_3\|= \sqrt{ 4}\ $ Let $w$ be a vector in ...
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2answers
28 views

proving that for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ (verify)

for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ I wasn't sure how to go about this correctly so what I did was set $v+w$ to $v$, yielding $w = v-v = 0$, since it ...
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Range of a function composition is a subset of the range [duplicate]

Let $L:\Bbb R^n → \Bbb R^m$ and $M:\Bbb R^m → \Bbb R^p$ be linear mappings. Prove that $Range (M◦L)$ is a subspace of $Range (M)$. So I began by defining: $Range (M◦L)$ is a subset of $\Bbb R^p$ ...
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Knowing if spans overlap

Only the first checked squares are deemed to be correct. Why is D not correct? After all, the vectors do overlap on the same plane...
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28 views

What are the different methods to determine if A is diagonalizable?

It seems every answer to finding out of a matrix is diagonalizable has a different approach. Where are all these different approaches derived from?
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35 views

Showing that $Det(A^T A)\ge 0$

How would it be shown this is property of determinants. Showing that $Det(A^T A)\ge 0$ My reason: It is know that $Det(A^T)=det(A)$ and by the multiplicative property it is know $Det(A^T ...
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1answer
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Is $Ape_1+Aqe_2$ where A (3x3) matrix, considered as a linear combination of $e_1,e_2$

$$\alpha=-8$$ Eigenvectors: $$e_1 = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} \text{ and } e_2 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$$ What I did : (i) $x ∈ V \implies x$ of the ...
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Checking if transformation T(p(x)) is diagonalizable?

Say you have a transformation of $P_3$ to $P_3$ defined by, say, $T(p(x)) = p'(x) + p''(x) + p'''(x)$. How would you determine if this is diagonalizable? Do I sub in a standard basis of ...
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61 views

Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is such a diagonal ...
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What is it called when we interpolate a point INTO a grid…

I suspect there is a terminological mish-mash going on in my understanding here: Consider a uniform 2D grid, where each $(x,y)$ value on this grid has a corresponding value. So, if I want to find ...
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46 views

If the inner product of Ax with x is 0 for all x, then A=0. [duplicate]

Given matrix $A\in M_{n}(\mathbb{C})$, if $\left<Ax,x\right>=0$ for all $x\in \mathbb{C^n}$, then $A=0_{n}$. (Here $\left<a,b\right> = b^{\ast}a$ where "*" is the conjugate transpose.) ...
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If $A$ and $B$ are real matrices and $X,Y$ are are non-singular square matrices such that $XA=BY$

If $A$ and $B$ are real matrices and $X,Y$ are are non-singular square matrices with real entries such that $XA=BY$ then which of the following is true? $1. \dim(X)=\dim(Y)$ $2. \dim(A)=\dim(B)$ ...
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Inspecting vector linear dependence, one line in matrix all zeros

Suppose we have vectors $v1, v2,v3$ and we want to inspect their linear dependence. They are linearly dependent when the only solution for the equation $\alpha * v1 + \beta * v2 + \gamma * v3 = 0$ is ...
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1answer
16 views

Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$

Let $V$ & $W$ be two finite dimensional vector spaces over $R$ and let $T_{1}:V\to V$ & $T_{2}:W\to W$ be two linear transformations whose minimal polynomials are given by ...
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1answer
20 views

Diagonalizable and Invertible Functions

In each of the following parts, either give an example of a linear function T: C$^2$ -> C$^2$ with the specified properties (and show that your example has the desired properties), or prove that no ...
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Convert global 2D coordinates to local

I have some 2D coordinates [x,y] and an object whose position and rotation is represented by a 3x3 matrix, with the form: [1 0 x] [0 1 y] [0 0 1] I need to get ...
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Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; $\forall n\in \mathbb{N}$

$a;b;c;x;y;z \in \mathbb{R}$ such that : \begin{matrix} a+b+c=x+y+z & \\ a^2+b^2+c^2=x^2+y^2+z^2 & \\ a^3+b^3+c^3=x^3+y^3+z^3 & \end{matrix} Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; ...
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Find a basis for a subspace of an inner product

Consider the vector space P$_2$(C), with inner product defined by $\langle{p(x)}$,$q(x)\rangle$ = $\int_0^1{p(x)\overline{q(x)}}dx$ Let W = {p(x) $\in$ P$_2$(C) : p'(0) = 0}. You may assume, without ...
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Determine whether the set of vectors is linearly dependent or not

Suppose I have the vectors $\underline{a}_1, \underline{a}_2,\ldots,\underline{a}_k$ and $\underline{b} \neq 0$ in $\mathbf{R}^n$. Also, $\underline{a}_1 \neq \underline{a}_2 \neq \ldots\neq ...
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Why is $P_{E^\bot}(x)=0$, if $x\in E$?

Let $H$ be a Hilbert space and let $x\in H$. Let $E$ be a non-empty closed subspace of $H$. Let $P_E(x)$ be the projection of $x$ unto $E$. I've seen several proofs that use the following: ...
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Intersection of kernel of commuting nilpotent matrices

Suppose $N$ and $Q$ are two nilpotent matrices which commute. Is it true that $\ker N \cap \ker Q \ne \{ 0\}$?
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Prove the following equation is solvable in $M_m(\mathbb C)$.

Prove the following equation is solvable in $M_m(\mathbb C)$ for any $n, l, m \in \mathbb N^*$: $${X^n}+{X^l}-{I_m}=\left( \begin{matrix} 1 & 0 & 0 & \cdots & 0 \\ 2 & 1 ...
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Proving that the sum of elements of two bases is a basis

I am given a (not necessarily orthonormal) basis of a certain finite vector space $\{e_i\}_{1\leq i\leq n}$. Now, after the usual Gram-Schmidt orthonormalization procedure, I end up with an ...
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Finding the conditions of a system of equations for a type of solution

Consider the system of equations $x$,$y$, and $z$, $$2x+3y-z=p$$ $$x-2z=-5$$ $$qx+9y+5z=8$$ where $p$ and $q$ are real. Find the values of $p$ and $q$ for which this system has: ...
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Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?

It is known that any infinite cyclic group , in particular $(\mathbb Z, +)$ , can never be a vector space . So we may ask , Can we define an operation $*$ on $\mathbb Z$ such that $(\mathbb Z , *)$ ...
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show that $A(t)\exp(\int_{t_0}^t A(s)\,ds )=\left(\exp(\int_{t_0}^t A(s)\,ds )\right)A(t)$, when $A(t)$ is symetric.

$A(t)$ is a symetric matrix for $t\in [t_0,a]$. show that $$A(t)\cdot \exp\left(\int_{t_0}^t A(s)ds \right)=\exp\left(\int_{t_0}^t A(s)ds \right)\cdot A(t)$$ it is easy but exhausting to show for ...
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Prove that a matrix is invertible?

Let $A_{20 \times 20}$ be a real matrix such that: $\ \ \ \bullet$ $a_{ii}=0$ for $1 \le i \le 20$ $\ \ \ \bullet$ $a_{ij} \in \{-1;1\}$ for $1 \le i,j \le 20$ and $ i \neq j$ Prove that $A$ is ...
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Linear Algebra, Spans and subspaces

Let $V= \mathbb{R^3}$ and consider the following elements of $V$: $\mathbf{u}_1 =(1,2,0)$, $\mathbf{u}_2=(3,1,0)$, $\mathbf{u}_3=(1,-1,1)$. Let $U= \langle\mathbf{u}_1,\mathbf{u}_2\rangle$ and ...
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The number of the solutions of $‎ x^{10}=‎ ‎ ‎\begin{bmatrix}1&0\\‎ ‎0&1‎ ‎\end{bmatrix}‎$

How many solutions does the following equation have in $ M_{2}(\mathbb R)$ and why? $$‎ x^{10}=‎ ‎\begin{bmatrix}1&0\\‎ ‎0&1‎ ‎\end{bmatrix}‎$$ Every hint is appreciated.
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Characterizing $\text{PGL}_2(\mathbb F_p)$

Where can I find a description and proof of the basic structure of $\text{PGL}_2(\mathbb{F}_p)$ (number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
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10 views

Singular Jacobian in Newton's method

How can we prove that Newton's method for a non-linear system converges linearly (as opposed to quadratically) if the Jacobian is singular at the root? Is this related to being multiple roots at that ...
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Can the zero vector be an eigenvector for a matrix?

I was checking over my work on WolfRamAlpha, and it says one of my eigenvalues (this one with multiplicity 2), has an eigenvector of (0,0,0). How can the zero vector be an eigenvector?
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39 views

Can a $3\times3$ matrix have more than $3$ linear independent eigenvectors?

I understand you can do multiples of eigenvectors, but suppose they are a linear independent. Can there be more than $n$ for a $n\times n$ matrix?
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Parametric / vector question.

Question 10 [10 points] Let L be the line with parametric equations $$ x = −6−3t $$ $$ y = 6+3t $$ $$ z = −8+2t $$ Find the vector equation for a line that passes through the point P=(−1, 2, 3) and ...