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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$\vec{u}+\vec{v}-\vec{w},\;\vec{u}-\vec{v}+\vec{w},\;-\vec{u}+\vec{v}+\vec{w} $ are linearly independent if and only if $\vec{u},\vec{v},\vec{w}$ are

I'm consufed: how can I prove that $$\vec{u} + \vec{v} - \vec{w} , \qquad \vec{u} - \vec{v} + \vec{w},\qquad - \vec{u} + \vec{v} + \vec{w} $$ are linearly independent vectors if, and only if ...
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1answer
76 views

Is this definition of diagonal matrix correct?

I need to know if the following definition: Let $A:=\|a_{i,j}\|_{\substack{i=1,...,m \\ j=1,...,m}}$ be a square matrix. $A$ is diagonal matrix if $$i\neq j \implies a_{ij}=0, \quad\forall i,j \in ...
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248 views

Proving when Gram's determinant is equal to zero [duplicate]

Prove that Gram's determinant $G(x_1,\dots, x_n)=0$ if and only if $x_1, \dots, x_k$ are linearly dependent. So I know that $G(x_1,\dots, x_n)=\det \begin{vmatrix} \xi( x_1,x_1) & \xi( ...
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1answer
87 views

Spectral radius of $A$ and convergence of $A^k$

I'm trying to understand the proof of first theorem here. Maybe it's very simple but I would like your help because I need understand this, I have no much time and my knowledge about this subject is ...
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1answer
300 views

Levenberg-Marquardt - Is forcing Hessian to be positive definite OK?

I am often doing parameter estimation using Levenberg-Marquard method which involves solving the following linear system at each step: $$(H+\lambda I)\delta=r_{i}$$ where $H$ is a square Hessian ...
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1answer
37 views

A linear map of the plane which preserves linear independence over Q but not over R

This is directly from Stewart & Tall's Algebraic Number Theory & FLT, chapter 8, Exercise 6, page 150. I can construct maps which do weird things but haven't been able to do this. Matrix ...
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82 views

about transpose of matrix

Let $A$ be a real $n\times n$ matrix with $A^{T}=\alpha_{0}I+\alpha_{1}A$, where $\alpha_{0}$ and $\alpha_{1}$ are real numbers. Show that either $A^{T}=\pm A$ or $A=\lambda I$ for some real number ...
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912 views

Prove that $T$ is an orthogonal projection

Let $T$ be a linear operator on a finite-dimensional inner product space $V$. Suppose that $T$ is a projection such that $\|T(x)\| \le \|x\|$ for $x \in V$. Prove that $T$ is an orthogonal projection. ...
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262 views

A strictly positive operator is invertible

Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is ...
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60 views

Verification of normality of linear operator

Let $X = (1,1,0),(1,-1,0),(0,0,1)$ be the basis of unitary space $\mathbb{C}^3$. Let $A$ be a linear operator, and $\mathbb{A}$ it's matrix in basis $X$: $$ \mathbb{A} = \begin{pmatrix}3 & i ...
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49 views

Jordan block: where (not) to place “one”?

Suppose we have some $\lambda$ with algebraic multiplicity of $4$ and geometric multiplicity of $2$. As I understand, the Jordan block will be $4\times 4$ matrix and $2$ "ones" on the subdiagonal. The ...
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2answers
74 views

Operators such that $T\circ S=I$ but $S\circ T\neq I$.

Suppose $S$ and $T$ are linear operators on a vector space $V$ and $T\circ S=I$ where $I$ is the identity map. It's easy to see that $S$ is one-to-one. If $V$ is finite dimensional, rank-nullity ...
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48 views

Eigenvalues of the real part of a complex matrix

Let $A\in M_{n}(\mathbb{C})$ and let $\{\lambda_{1},...,\lambda_{n}\}$ be the eigenvalues of $A$. Is it true that the eigenvalues $\{\mu_{1},...,\mu_{n}\}$ of $\frac{A+\bar{A}}{2}$ are of the form ...
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39 views

about the rank of block matrix

Let $m,n,$ and $k$ be positive integers and assume that $A\in {\mathbb{R}^{m\times n}}$,$B\in {\mathbb{R}^{k\times n}}$,( that is, A and B are matrices with real entries of sizes $m\times n$ and ...
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1answer
45 views

find X, such that U=VX

Let $\{u_{1},u_{2},\ldots,u_{n}\}$, $\{v_{1},v_{2},\ldots,v_{n}\}$ be two bases for $\mathbb{R}^{n}$, and define $U=[u_{1}\mid u_{2}\mid\cdots\mid u_{n}]$, (that is, $U$ is the $n\times n$ matrix ...
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1answer
163 views

invariant space and eigenvector

Let $V$ be a finite-dimensional vector space over $C$, let $T:V\rightarrow V$ be a linear operator, and let $M$ be a nontrivial subspace of $V$ that is invariant under $T$. Prove that $M$ contains an ...
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2answers
112 views

span of $AA^T$ is the same as $A$?

Suppose $A$ is an $m$ by $n$ real matrix. How do you prove that the span of columns of $AA^T$ is the same as columns of $A$?
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86 views

Range(A) and Columns

let $A$ be a $m\times n$ matrix. Show that (a) $Null(A)=\{0\}$ if and only if the columns are linearly independent. (b)$Range(A)=\mathbb{R}^{m}$ if and only if the columns span $\mathbb{R}^{n}$. I ...
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132 views

Linear Algebra and Fredholm's Alternative Problem

Let $A=$ $ \left[ \begin{array}{ccc} 1 & -3 & 2 \\ 2 & 1 & -1 \\ 3 & -2 & 1 \end{array} \right]$ (a) Determine the value of $\beta$ such that $Ax=\left[\begin{array}{ccc} 4\\ ...
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51 views

Conditioning on $X$ equal to premultiplying by $X'$?

I am coming across similar thing in many problems in econometrics and I have not been able to figure out whether it is some general notion or only a "coincidence". To take two examples: Deriving ...
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1answer
78 views

Find the best fit (left best inverse) for a given function

find the best fit (left best inverse) for the equation: $$y = \alpha_0 + \alpha_1 x + \alpha_2 x^2 + \alpha_3 x^3$$ given the data points $(x,y)$: $(-5,-5002) (-3,-1100) (0,2) (3,570) (5,4780)$ ...
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What is the use of the Dot Product of two vectors?

Suppose you have two vectors a and b that you want to take the dot product of, now this is done quite simply by taking each corresponding coordinate of each vector, multiplying them and then adding ...
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2answers
103 views

rank of a diagonal matrix + rank-one perturbation

Let $D$ be a $n \times n$ diagonal matrix, and $A$ is a $n \times n$ rank-one matrix that can be rewritten as $A=a\cdot b^T$, where $a$ and $b$ are $n \times 1$ vectors. Now what is the lower bound ...
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1answer
44 views

Prove that $E_1E_2= E_2E_1 = E_2$

I have this problem about projections I don't understand, Can somebody help me please? Let $V$ be a vector space over the field $F$ and let $E_1$ and $E_2$ are projections of V with image $R_1$ and ...
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2answers
739 views

Eigenvectors of inverse complex matrix

For a non-singular matrix, its pretty straightforward to prove that $\lambda$ is eigenvalue of $A$ if and only if $\frac{1}{\lambda}$ is eigenvalue of $A^{-1}$. Let $A$ be a non-singular matrix, $x$ ...
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1answer
392 views

Difference between convergence in norm, point-wise and uniform convergence

I know both definitions but I was wondering what are the relations between them. My question is if someone could explain intuitively the differences between these types of convergence. Specifically, ...
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164 views

Spectral radius and operator norm

Consider a FINITE endomorphism $A$ , then I was wondering whether the relation between the operator norm and the spectral radius $\rho$, given by: $||A|| \ge \rho(A)$ is true for all operator norms or ...
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rth adjugate matrix and determinant of a sum, in Horn

In Horn, matrix analysis, 2nd edition, it is stated p.29 that $$\det (sA+tB) = \sum_{k=0}^{n} s^kt^{n-k}\text{tr}(adj_k(A)C_r(B))$$ $adj$ is the $r-$th adjugate matrix, $C_r$ is the $r-$th compound ...
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1answer
397 views

Thin Plate Spline interpolation of scattered $z(x,y)$ data

I am trying to understand Thin Plate Spline interpolation of scattered data. As I understand it TPS is just a special case of Radial Basis Function interpolation: $$ z(x,y) = p(x,y) + \sum_i ...
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115 views

Every orthogonal operator is diagonalizable?

Answer is false and the rotation is a counterexample. But I can't understand well. Let $A=\begin {pmatrix}0&-1\\1&0\end{pmatrix}$ then it is rotation and also orthogonal operator. I think it ...
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42 views

Angle between matrices

This is the problem from my homework: If $A$ is antisymmetric matrix, and $S$ is symmetric matrix where $A,S \in M_n (\mathbb{R})$, determine the angle between them according to the inner product ...
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1answer
298 views

proof about cyclic vectors and irreducible polynomial

Let $V\neq \{0\}$ be a vector space over $F$, and $T$ a linear operator on $V$. Prove that every $0\neq v \in V$ is a cyclic vector if and only if the charictaristic polynomial of $T$ is irreducible ...
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43 views

Computing intersection of two subspaces of $C^{\infty}_{2\pi}(\mathbb{R},\mathbb{R})$.

I've been thinking of the following two subspaces of $C^{\infty}_{2\pi}(\mathbb{R},\mathbb{R})$: $$ A=\{a_1\sin(t)+a_2\sin(2t)+a_3\sin(3t):a_1,a_2,a_3\in\mathbb{R}\} $$ and $$ ...
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1answer
167 views

Multiplying vectors (answered own question)

I recently realised that asking a question and answering our own question is allowed here, so here is a question I've seen commonly on many sites: "How does one multiply two vectors?" This is very ...
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1answer
91 views

Singular value decomposition problem

Let $A\in \mathbb{R}^{m\times n}$ have singular values $\sigma_{1}\ge\sigma_{2}\ge\cdots\ge\sigma_{k}$ (where $k=\min\{m,n\}$).Assume $\sigma_{r}>0$, $\sigma_{r+1}=0$. Prove that ...
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2answers
178 views

Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
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3answers
281 views

Why Gaussian Elimination only works over field?

When I was solving system of linear congruences (n variables, n equations), like this: $AX \equiv b \pmod p$ I was told that ordinary Gaussian Elimination works if $p$ is prime. And I figured out ...
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111 views

Question about inner product.

Let $V=C([-1, 1])$ and $$\langle f, g\rangle=\int_{-1}^1 f(x)g(x)dx$$ Let $W=\{f \in V \mid f\text{ is even}\}$. Find $W^\perp$. Progress: I know that every odd function belongs to $W^\perp$ and I ...
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91 views

For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?

I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you ...
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461 views

How many Jordan normal forms are there for this characteristic polynomial?

Given the characteristic polynomial of a matrix $A \in \mathbb{C}^{6x6}$ with $p(A)=(\lambda-2)^2(\lambda-1)^4$, we were supposed to determine all Jordan normal forms that have this characteristic ...
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84 views

Strange exercise Jordan normal form

I have the following excercise that I cannot answer: Imagine that you have measured two entries $\tilde{A}_{11},\tilde{A}_{22}$ of the matrix $\tilde{A}=\left(\begin{smallmatrix}\\1+\epsilon & 1 ...
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177 views

Is $f'(x)-3f(x) = 0$ subspace of differentiable functions $f\colon (0,1)\to \mathbb{R}$

$V$ is space of differentiable functions $f(0,1) \to \mathbb{R}$ and $W$ is a subset of $f$ that meets $f'(x) - 3f(x) = 0$ for all $x\in (0,1).$ Is subset $W$ a subspace of $V$? I know that I have ...
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1answer
107 views

Question about inner product

Question: Let $V$ be a finite dimensional vector space with $\langle\ ,\, \rangle$ inner product and let $T$ be a linear operator on $V$. Suppose $\|u\| = \|v\| \implies \|Tu\| = \|Tv\|$ $\forall ...
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96 views

Eigenvectors from eigenvalues doesn't add up

I having some trouble understanding how to find eigenvectors of a matrix. I have the following matrix: $A=a\left[ \begin{matrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & ...
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53 views

Finding Jordan form of a matrix

Is there a way to know the original matrix when given its Jordan form? For example, the Jordan form of matrix $A$ is $\text{diag}(J_3(1), J_3(0))$, so I know that the $Pa(x) = x^3(x-1)^3$. I need ...
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7answers
494 views

How to find a nonzero $2 \times 2$ matrix whose square is zero?

How do I find a nonzero $2 \times 2$ matrix $A$ such that $A^2$ has all zeros entries? Very confused with this. Could use all the help I can get. Thank you
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281 views

proof about commutative operators and T-cyclic vectors

Let $V$ be a finite dimensional vector space over $F$. Let $T:V \to V$ be a linear operator. Prove that if every linear operator $U$ which commutes with $T$ is a polynomial of $T$, than $T$ has a ...
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2answers
463 views

Why is it linearly dependent when the linear combination is zero only with none zero coefficients in 3D?

Title says it all. I'm asking the geometrical sense. I know it is linearly independent if the linear combination of vectors is zero with all the coefficients are zero. And so do dependent. ...
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556 views

Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?

I'm a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a square into a quadrilateral (as ...
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1answer
79 views

Translation matrices exercices [ check my answer please ]

Let $L:R^2 \to R^3$ be linear transformation with equation: $L((x_1,x_2)^T) = (x_2,\ x_1 + x_2,\ x_1 - x_2)^T$ Let: $ u_1 = (1,2)^T, u_2 = (3,1)^T $ and $b_1 = (1,0,0)^T, b_2 = (1,1,0)^T, b_3 = ...