Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Solving a matrix equation

Let $\alpha \in \mathbb{R}$, $(\,u \ \ v \ \ w\,)^\textrm{t} \in \mathbb{R}^3$, $S = S^\textrm{t} \in \mathbb{R}^{3 \times 3}$ with $$\alpha + (\,u \ \ v \ \ w\,)\,S\, \left(\begin{matrix}u \\ v \\ ...
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27 views

When can a matrix be written as $UDU^H$

I have read a solution to the problem of diagonalizing a matrix $A$. The solution started with the observation "note that $A^H=A$ thus we can diagnalize like this $A=UDU^H$ where $U$ is a unitary ...
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1answer
29 views

Showing a reduction by 30% using dimensional analysis

I have used dimensional analysis to show that Energy needed to overcome drag at constant speed is $E = C \times v^2 \times A \times p$ where C is a constant $v$ is the constant speed, $A$ is surface ...
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42 views

Necessary and sufficient condition for linear transformation having charactristic 2 to be self-inverse

Is there any necessary and sufficient condition for linear transformation on the vector space having characteristic 2 to be self-inverse?
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30 views

Notion of density in matrices

Is there a formal way of defining a "dense" matrix? What are some properties of such matrices? I suspect it has something to do with how "easy" it will be to diagonalize them; if so, is there a way to ...
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2answers
81 views

What does this theorem mean?

Let $(V,\|\cdot\|)$ be a finite-dimensional normed space. Define $\|T\|_\mathrm{op}=\sup\{\|T(x)\|:\|x\|≦1\}$, for all linear operators on $V$ Define $\Omega$ to be the set of all invertible linear ...
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2answers
106 views

Find the 3x3 matrix which the system is inconsistent.

Find the value of K such that the system is inconsistent. $$\begin{cases}x + y + 2z = 7\\ -2x - 2y + Kz = -14\\ 3x + 3y + 6z = 14\end{cases}$$ Thanks for the help...
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41 views

Linear transformations relate to operator algebra

$P(t)$ is a polynomial. $A$ is a linear transformation on $P$. $P(t) = a_0 +a_1t +a_2 t^2+\cdots + a_nt^n \qquad P(A) = a_0 + a_1A + a_2 A^2+\cdots+ a_n A^n$ I do not understand how t and A relate ...
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38 views

$T$ is a linear operator on a IPS $V$ which has a basis $\beta$. Prove that $A_{ij} = \langle T(v_j),v_i \rangle$

I have trouble understanding a proof on textbook and I would appreciate your help! Corollary. Let $V$ be a finite-dimensional inner product space with an orthonormal basis $\beta = \{v_1, v_2, ...
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29 views

A System of Matrix Equations

I am looking for the solutions to the following system of matrix equations: (1) $AA^{*} - A^{*}A = C^{*}C - BB^{*}$ (2) $DD^{*} - D^{*}D = B^{*}B - CC^{*}$ (3) $AC^{*} - C^{*}D = A^{*}B - ...
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Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
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2answers
601 views

Show that the four points given below are the vertices of a rhombus.

Show that the four points, $(5, 8), (7, 5), (3, 5)$ and $(5, 2)$ are the vertices of a rhombus. I tried solving it, by finding out the distances by using the formula $\sqrt{(x_{2}-x_{1})^2 + ...
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1answer
125 views

$F,G \in \text{End} (V)$ share the same eigenvalues for $F \circ G$ and $G \circ F$

Problem: Let $V$ be a finite dimensional Vector Space over a field $\mathbb{F}$ and $F,G \in \text{End}(V) $ Show that $F \circ G$ and $G \circ F$ have the same Eigenvalues $\lambda$ My ...
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45 views

Check this problem of linear algebra why this happened

showing the subspaces spanned by$S=\{\alpha,\beta\} $ and $T=\{\alpha,\beta,\gamma\}$ where $\alpha=(1,2,1),\beta=(3,1,5),\gamma=(3,-4,7)$ , I got $\gamma=-3\alpha+2\beta$ and here both $S\subseteq T$ ...
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1answer
44 views

How to solve an Optimization problem with linear as well as Quadratic constraints.

I want to solve the following problem, \begin{equation} \begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & \mathbf{x^T}\mathbf{Px} \\ & \text{subject to} & & ...
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87 views

Properties of the evaluation functional of a space of polynomials

Let $K$ be a field. Let $E=K_{n}[x] = \{p(x) \in K[x] : degree(p(x)) \le n\}$. (a) Given $a \in K$, show that the lineat transformation $\omega: E \rightarrow K $ defined by $\omega(p(x))=p(a)$ it's ...
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41 views

How to rearrange this

Given $$x^2+\frac{2D}{k}x=\frac{2DC_{o}}{C_{l}}(t+\tau)$$ How do I get to $$x=\frac{D}{k}\left[\sqrt{1+\frac{2C_{o}k^2(t+\tau)}{DC_{l}}}-1\right]$$ Thanks!
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60 views

Condition for a particular matrix to be semi definite positive

Let $B$ a symmetric $N\times N$ real matrix whose diagonal elements are equal to one, that is to say $B_{i,i}=1$, $\forall i = 1, \dots N$ $B_{i,j} = B_{j,i}$, $\forall i,j = 1, \dots N$. $B_{i,j} ...
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287 views

need orthogonal basis for R3. i'm given one of them. how do i find the rest?

I need orthogonal basis for R3. I am given v1 = (1,1,1), so I need so I need other two vectors in this basis but how do i find the other two? at first i thought i would use gram schmidt but that ...
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5answers
93 views

General form for $A^2=0$ where $A$ is a $2\times2$ matrix [closed]

Find the general form for all the $2\times2$ matrices over $\mathbb{R}$ that $A^2=0$
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Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices

The component-wise product (Hadamard product) of two positive definite matrices is a positive definite matrix (Schur product theorem). I encountered the following proof of it: $A=(a_{ij})$ and ...
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102 views

linear Combination And The Zero Vector

I came across the notion that every linear combination with zero is linear dependent because every scalar that is not zero can be attach to it. So that mean that every space with the zero vector must ...
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2answers
168 views

Trace of the power matrix is null

Let $K$ be a field of characteristic $p \geq 0$ and let $M$ be a matrix $n \times n$ over $K$. If $p \nmid n$ and $Tr(M^i) = 0$ for all $i = 1,\dots,n$, how to prove that $M + Id_n$ is invertible? If ...
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165 views

Linear Algebra Self Study

I'm currently a high school student with a love for math. I have taken Plane and Coordinate Geometry, both Algebra I and II, Trigonometry, and am halfway done with Calc A. I want to major in quantum ...
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65 views

Linear Algebra quick Question over inner product space

In an inner product space, not necessarily $\mathbb R^n$, there are vectors $a$ and $b$ such that $||a||\cdot ||b|| < |\langle a,b\rangle| $ Is this never true?
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124 views

Linear Algebra Explanations on true and false.

1.Could someone prove that if a set of vectors in a $p$-dimensional vector space $Q$ is a spanning set for $Q$, it is a basis. 2.If $T$ is a linear transformation from $\mathbb R^3$ onto $P_2$, then ...
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2answers
79 views

Linear Algebra Don't Understand a question involving basis

the question is Find $[t^2-3t+4]_B$ I am given a basis $B=\{4,1+t,1-t^2\}$ Could someone explain how to attempt this problem and what exactly am I supposed to do? I am not asking for the answer.
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Linear Algebra question on diagonlization Please check my work

My first question is that is a basis for each eigenspace the same thing as a corresponding eigenvector for an eigenspace? Could someone tell me if im doing this correctly? I have the matrix ...
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1answer
292 views

Volume of ellipsoid using Linear Algebra

Can someone tell me how to find the volume of an ellipsoid of dimension $\mathbb{R}^3$ by using linear algebra? I know the formula is $\frac{4}{3}\pi abc$. I am given the equation ...
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1answer
54 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
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1answer
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Linear transformation and eigens

Hi guys I need some help understanding how find Eigenvectors for the following problem. Define $T \in \mathscr{L}(F^2)$ by $T(w,z) = (z,w)$ So I set up the problem like so... $z=\lambda w $ ...
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Number of ellipses to uniquely define a co-centered circumscribing ellipse

I have a bit of a tricky problem that has come up in my engineering research, but I haven't quite got the brains to figure it out, though I've gotten pretty far. Suppose that there is an unknown ...
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1answer
46 views

Is an algebraic equation is a form of algebraic expression or are they different?

The difference between Algebraic expression to algebraic equation is the presence of equal sign. But can we say that an algebraic equation is also a form of algebraic expression since it also ...
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5answers
215 views

Prove If $A^{2014}$ is invertible, then $A$ is also invertible

Use the associativity of matrix multiplication to prove that if $A^{2014}$ is invertible, then $A$ is also invertible. Any help please?
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Find the equation of the line perpendicular to AB passing through B

I'm not sure on how to work out this question: 'Find the equation of the line perpendicular to AB passing through B ...
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2answers
589 views

show projection matrix is equal to matrix times its transpose

Let $V$ be an $n$-dimensional real inner product space and let $a=\lbrace v_1,v_2,\dots v_n \rbrace$ be an orthonormal basis for $V$. Let $W$ be a subspace of $V$ with orthonormal basis $B = \lbrace ...
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1answer
88 views

Prove that for every subspace we can find a finite number of linear functionals such that $W=\ker l_{1}\cap\cdots\cap \ker l_{k}$

In need of some assistance regarding this questions from a University textbook (I'm learning by myself). Its about Dual Spaces: Let there be $V$ a finite vector space (Has a basis) over $\mathbb{F}$. ...
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Help with Dual Spaces - Prove that either $w\in Im(f)$ **or** there exists ${l\in W^{*}}$ such that $f^{*}\left(l\right)=0$ **and** $l(w)=1$"

I'm in need of some assistance regarding this question. I'm learning Linear Algebra by myself using a university textbook and it has this question regarding Dual Spaces: "Let there be a linear map ...
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1answer
54 views

Understanding a Proof for Why $\ell^2$ is Complete

Setting: Let $(x_n)$ be Cauchy in $\ell^2$ over $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. I'm trying to show that $(x_n) \rightarrow x \in \ell^2$. That is, I'm trying to show that $\ell^2$ is ...
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1answer
44 views

mathematical induction on product of matrices

Let $M_1,\dots, M_k (k\geqslant2)$ be matrices and their orders may not be the same. However, $M_1M_2\cdots M_k$ is a product of matrices which is a well defined square matrix. Prove that ...
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334 views

Matrix multiplication as combination of rotation and stretching

I'm starting to look at some lectures on the SVD. The lectures start out by saying that in $$\mathbf{y}=A\mathbf{x}$$ the transformation representing $A$ is only doing rotation and stretching. The ...
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1answer
613 views

How can I use math to fill out my NCAA tournament bracket?

With the NCAA basketball tournament right around the corner and the conference tournaments just beginning, it's the perfect time to consider strategies to fill out an NCAA tournament bracket. How can ...
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1answer
54 views

Proof of well-known principle used in the investment industry for compliance criteria

I work in the financial services industry. Presently I am involved in more than one project involving the monitoring and reporting on compliance of investment portfolios. A principle that is ...
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Particular determinant made of powers of algebraic numbers is nonzero?

Let $P$ be a degree-two polynomial, with roots $\alpha,\beta$. Is there a simple condition on $P$ (or on $\alpha,\beta$), equivalent to the following : $$ ...
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3answers
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Find a second degree polynomial that goes through 3 points

I am having trouble calculating the quadratic curve $f(x)$ that goes through 3 points; for example a curve that goes through $A(1,3), B(-1,-5), and C(-2,12)$. I can only guess that the curve is ...
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0answers
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Basis and their orientation

let V be a vectorspace with $v_1 = (3,2,1), v_2 = (2,2,1), v_3 = (1,1,1)$. Do the two basis $A = (v_1, v_2, v_3)$ and $B = (v_2, v_3, v_1)$ have the same orientation? Since this is a new thematic for ...
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Show that $T$ is a linear transformation given Orthonormal basis

Suppose that $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and suppose that $\{v_1,v_2,\cdots,v_n\}$ and $\{Tv_1,Tv_2,\cdots,Tv_n\}$ are orthonormal basis of $\mathbb{R}^n$. Prove that $T$ is a linear ...
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78 views

Is this easy matrix diagonalizable or not?

Question Is $$ \pmatrix{ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 } $$ diagonalizable if we allow only real numbers? if we ...
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1answer
303 views

Matrix rotation, projection, and reflection

What 3 by 3 matrices represent the transformations that a) project every vector onto the $x-y$ plane? b) reflect every vector through the $x-y$ plane? c) rotate the $x-y$ plane through 90 degrees, ...
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2answers
95 views

Inverse of Triangular matrix [duplicate]

I want to know if the inverse of a Triangular matrix (Cholesky decomposition of a symmetric, positive-definite matrix in my particular case) is still triangular. If so could someone provide a proof as ...