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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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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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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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

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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26 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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1answer
34 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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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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1answer
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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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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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1answer
43 views

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

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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1answer
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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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1answer
14 views

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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1answer
21 views

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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1answer
15 views

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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1answer
32 views

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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1answer
26 views

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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1answer
25 views

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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1answer
26 views

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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3answers
78 views

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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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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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 ...
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A basic doubt to compute exponential of a matrix

Given a matrix I want to evaluate $e^{A}$. The method suggested uses the taylor expansion. But, it is also written that the method works well if the largest and smallest eigen values are not well ...
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1answer
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How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...
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Does this guassian elimination have a solution?

I was asked to find the following solutions using guassian elimination, but I was unsure of my answers since it became quite messy but the variables still somehow fit: $$\left[\begin{array}{ccc|c} ...
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25 views

The expected value of a random vector when the X_is are independent

$ \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cov}{cov} $ The components of a random vector $\mathbf{X} = [X_1, X_2, \ldots, X_N]^{\intercal}$ all have the same mean $E_X[X]$ and the same ...
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Prove True or false : If A and B are nxn invertible matrices and (AB)^2=A^2B^2, then AB=BA

This looks like it is false but the thing is I can't find a counter example for it.
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Fixed field of two subgroups of $\operatorname{Aut}_{K}{K(x)}$

This link explains $\operatorname{Aut}_{K}{K(x)}$. And I want to know how to solve two problems below in the Hungerford's Algebra, p.256. $7.$ Let $G$ be the subset of $\operatorname{Aut}_{K}{K(x)}$ ...
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Matrix Norm Division

Suppose $A=uv^*$ where $u$ is an $m$-vector and $v$ is an $n$-vector. For any $n$- vector $x$, we can bound $||Ax||_2$ as follows: $||Ax||_2 = ||uv^*x||_2=||u||_2|v^*x|\leq||u_2||||v||_2||x||_2$. ...
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1answer
21 views

Number of positive, negative eigenvalues and the number of sign changes in the determinants of the upper left submatrices of a symmetric matrix.

How do we prove that the number of sign changes in the sequence of the determinants of the upper-left matrices of a symmetric matrix $A$ corresponds to the number of positive and negative eigenvalues ...
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24 views

linear algebra find a line that intersects another line

question: Let L be the line with parametric equations x = 3+2t y = −5 z = −6−3t Find the vector equation for a line that passes through the point P=(−5, 5, −6) and intersects L at a point that is ...
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2answers
30 views

Sets of binary sequences

In my course on linear algebra we have recently introduced linear independent subsets of vector spaces. As an exercise I have been thinking about examples of infinite linearly independent sets and ...
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32 views

Does basis of eigenspace mean the same as eigenvectors?

If you have a 3x3 matrix, 2 eigenvalues (one with multiplicity 2) and now 2 eigenvectors, how do you find the basis for each eigenspace?
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Orthogonal Matrices and Similarity Transforms

Sorry I can't be more specific with the title. I really don't know what to call this and about 2 hours of Googling has yielded no results. All we are given: $U$ is $n\times n$ and orthogonal $Ax = ...