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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Linear algebra: Matrix identity

I am trying to understand a derivation and there is a matrix manipulation which I do not understand. So, there is the following derivative: $$ \frac{d}{dx} (x^T\Sigma^{-1}x) $$ Here x is a D ...
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Geometry question pertaining to $4$ points in the plane where $90$ degree projectors are on each point and we must illuminate the whole plane.

Suppose we have $4$ points that can be positioned anywhere in $\mathbb{R}^2$. Now imagine each point has $90$ degree projectors coming out of them and you can rotate these projectors any way you would ...
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Is T well-defined? Find the matrix representation.

Let $W_1$ be the subspace of C(0,1) spanned by the functions $\{e^x,xe^x,x^2e^x\}$. Let $W_2$ be the subspace of C(0,1) spanned by the functions $\{1,e^x,xe^x,x^2e^x\}$. Let T be the application ...
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Is an orthogonal projector bounded in $L_p$-spaces?

Let $P$ be an orthogonal projector on $L_2([0,1])$. In particular, $P$ is a projector from $C^\infty([0,1])$ to itself. For $0<p<\infty$, we define for $f \in C^\infty$ the norm (quasi-norm if ...
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Proof of an interesting matrix property

Suppose you have a square matrix $M$ with $n$ rows and $n$ columns. Suppose $M$ enjoys a property $p$ defined as follows: $M_{i,j} = 0$ if $i + j$ is odd and non zero otherwise. Question: if square ...
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4answers
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Show that $\operatorname{rank}A+\operatorname{rank}A^m \leq n$ where $A^{m+1}=0$

Let $A$ be an $n \times n$ nilpotent matrix over a field $F$. Show that $$\operatorname{rank}A+\operatorname{rank}A^m \leq n$$ where $A^{m+1}=0$. By FTLA, it is equivalent to $$ n \leq ...
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1answer
57 views

Property of normal matrix's eigenvalue

How do I prove that, if $A\in\mathbb{C}^{n\times n}$ is a normal complex matrix (i.e. $AA^H=A^HA$, $A^H$ being the conjugate transpose) and $f$ a rational complex function defined on a subset of the ...
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1answer
87 views

Function such that $\,f,f',\dots,f^{(n-1)}\,$ are linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$ such that $\,f,f',\dots,f^{(n-1)}\,$ are linearly independent while $\,f^{(n)}=f$. Could you give me some hints? I truly ...
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System of linear equation matrix? [duplicate]

How would I do this question. Determine the value(s) of $h$ such that the matrix is augmented of a consistent linear system. My matrix \begin{bmatrix} 1&h&4\\ 3&6&8 \end{bmatrix} I ...
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4answers
50 views

Show linear independence

Is the Set $$S=\{e^{2x},e^{3x}\}$$ linearly independent?? And answer says Linearly independent over any interval $(a,b)$,only when $0$ doesnot belong to $(a,b)$ How do I proceed?? Thanks for the ...
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1answer
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Representation of a linear functional

Let $f, f_{1}, . . . , f_{n}$ be linear functionals on a linear space $L$ such that $f_{1}(x) = . . . =f_{n}(x) = 0$ implies $f (x) = 0$. Prove that there exist constants $a_{1},, . . , a_{n}$ such ...
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1answer
18 views

Understanding the operator of differentiation on the vector space of polynomials

I have been reading through Linear Algebra Done Right by Sheldon Axler. The book defines an operator as a linear map from a vector space to itself. It then considers at another part of the book the ...
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2answers
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Necessary and sufficient conditions for when spectral radius equals the largest singular value.

One well known fact about matrix norms is the following: If $\lambda_1\geq \dots\geq \lambda_n$ are eigenvalues of a square matrix $A$, then: $$\frac{1}{||A^{-1}||} \leq |\lambda|\leq ||A||$$ If we ...
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Finding the Jordan canonical form of this upper triangule $3\times3$ matrix

I am supposed to find the Jordan canonical form of a couple of matrices, but I was absent for a few lectures. \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 3 ...
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2answers
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Jordan form of a Matrix with Ones over a Finite Field

Question: Find the Jordan Form of $n\times n$ matrix whose elements are all one, over the field $\Bbb Z_p$. I have found out that this matrix has a characteristic polynomial $x^{(n-1)}(x-n)$ and ...
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1answer
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Find the intersection between two linear equations where one is y=? and another is x=? [on hold]

I have two equations: y=4 x=3.58 Find the intersection between the two equations.
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1answer
27 views

(dis)proving $Span((\cos nx)_{n\in\mathbb{N}})=Span((\cos^nx)_{n\in\mathbb{N}})$ in $\mathbb{R}^\mathbb{R}$

I am trying to show that $Span((\cos nx)_{n\in\mathbb{N}})=Span((\cos^nx)_{n\in\mathbb{N}})$ in $\mathbb{R}^\mathbb{R}$. ($0\in\mathbb{N}$) I immediately thought of the Chebyshev polynomials : ...
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0answers
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Gradient calculation for Matrix

I have the following: $$b^T a^{-1} b$$ what is the gradient wrt to $a$. $a$ is matrix and $b$ is vector. Basically I should take the derivative with respect to $a$. Is it correct that it equals to: ...
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1answer
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About a proof of the minimal polynomial of matrix doesn't change when the field changed

In Keith. Conrad's lecture notes, he proved the following theorem theorem: Let $K/F$ be any field extension. (1) For any $A\in\operatorname{M}_n(F)$, its minimal polynomial in $F[x]$ is its ...
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1answer
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Calculate the distance from the centre to the edge of an ellipsoid

So, I'm aware that there is a formula to calculate the distance from center to edge of an ellipse. My problem, however, is in three dimensions. I can formulate it thusly (english is not my first ...
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1answer
27 views

Is the largest eigenvalue a unique weighted sum of the linear combination of the elements of a matrix?

Let $\lambda$ be the largest eigenvalue of $\boldsymbol{A}\in\mathbb{C}^{n\times n}$ ($\boldsymbol{A}$ is hermitian). Is $$\lambda = ...
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A power series of $2\times2$ matrices

Let A= $\begin{bmatrix} 0 & 1 \\ 1 & 0\ \end{bmatrix}$, $I$ is the identity matrix, what is $I+\sum_{n=1}^{\infty}\frac{t^n}{n!}A^n$?
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1answer
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Any segment can't be inside the sphere

I'm trying to prove if two points $a$ and $b$ are in the closed ball, then the segment between them is inside the ball, and can't be in the sphere, in another words: Let $a,b\in \mathbb R^n$, $a\neq ...
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2answers
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What is wrong with this proof that if $V = U_1 \oplus W$ and if $V = U_2 \oplus W$, then $U_1 = U_2$?

Claim: Let $U_1, U_2$ and $W$ be subspaces of a vector space $V$. Suppose $V = U_1 \oplus W$ and $V = U_2 \oplus W$. Then $U_1 = U_2$. "Proof" Let $v \in V$. Then $\exists \space u_1 \in U_1 $ ...
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Liner algebra by K.H Hoffman (orthogonal) [on hold]

Zero vector is orthogonal to every vector in V. And it is the only vector with that property.
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1answer
16 views

Fixed and Variable Cost Question

I have the following question: A restaurant has fixed costs that are \$34,000 per month. Its variable costs average \$1.80 for breakfast and \$3.70 for lunch/dinner. The average total bill ...
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0answers
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Matrix Inversions

I have the following problem: "Suppose $P$ is invertible and $A=PBP^{-1}$. Solve for $B$ in terms of $A$." As far as I can tell, the value of $B$ depends of the values of both $A$ and $P$, not just ...
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Use Schur complement theorem to transform an inequality into semidefinite form?

This problem is from the paper: B. K. Chalise, Y. D. Zhang, and M. G. Amin, “Energy harvesting in an OSTBC based amplify-and-forward MIMO relay system,” in Proc. IEEE ICASSP , Kyoto, Japan, Mar. 2012, ...
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Orthogonal Complement

"Let $\Bbb{V}$ be a vector space with an inner product $<\cdot,\cdot>$, and $S\subset\Bbb{V}$. We define the orthogonal complement of $S$, denoted by $S^{\perp}$, as follows: ...
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How are eigenvectors/eigenvalues and differential equations connected?

In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...
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Hyperplanes as dual projective spaces

I was reading through Harris's Algebraic Geometry book, and was slightly perplexed by the following paragraph: "Note that the set of hyperplanes in a projective space $\mathbb{P}^{n}$ is again a ...
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How to stop iteration in inverse problem using nonlinear least square problem?

I am having a real trouble with stopping criterion in iteration of Generalized Nonlinear Least Square. My problem is that I do not know exactly how to stop my iteration. First, I will give a short ...
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1answer
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$\{id_V,T,T^2,…,T^d\}$ is linearly independent

Let $V$ be a finite dimentional vector space over a field $F$, and $T:V\to V$ diagonalizable, where $c_1,...,c_r$ are the distinct eigenvalues of $T$. Prove that a) $p(t)=(t-c_1)\cdot ...\cdot ...
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Determining Distance between two objects known in Size and distance of one object

I have a 10 Cent in my Hand (Diameter 19,2mm) and a DIN-A4 Paper on the table (297mm) I am holding the coin in front of my eye so that it fills the Paper, and i am using the following formula: x = ...
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Find Homogeneous System from Solution Spaces

I have the following vectors in the Subspace of $\mathbb{R}^5$ $U=\mbox{Span}[(1,-1,-1,-2,0), (1,-2,-2,0,-3), (1,-1,-2,-2,1)]$ $W=\mbox{Span}[(1,-2,-3,0,-2), (1,-1,-3,2,-4), (1,-1,-2,2,-5)]$ I need ...
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Determinants in pairs of fundamental solutions to particular types of linear, time-varying ODEs

Consider a vector-valued ODE of the following form $$ x'(t) = \begin{bmatrix} 0 & A(t) \\ B(t) & 0 \end{bmatrix}x(t) = \Xi(t) x(t), $$ where $x(t) \in \mathbb{R}^{2n}$ and $A$ and $B$ are ...
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0answers
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Method for determining next pivoting row?

Given systems of linear equations like $A$, how does one computationally find out what order to place the rows to ensure that no diagonal elements become zero during Gaussian Elimination? (If the ...
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1answer
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Find the eigenvalues and eigenvectors of the linear transformation $T(x,y,z)=(x+y,x-y,x+z)$. Verify that the eigenvectors are orthogonal.

Find the eigenvalues and eigenvectors of the linear transformation $T(x,y,z)=(x+y,x-y,x+z)$. Verify that the eigenvectors are orthogonal. Part A: $$T(x,y,z)=\begin{pmatrix} 1 & 1 & 0 \\ 1 ...
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1answer
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A normal matrix with real eigenvalues is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
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If A is regular, then $AA^T$ is positive definite, since $x^TAA^Tx=(A^Tx)^T(A^Tx)>0$

I read this statement and didn't understand why the right part of the equation is true. Namely, that: $(A^Tx)^T(A^Tx)>0$ Can someone explain? Thank you.
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Projection from triangle to $\mathbb{R}^2$.

I constructed the $2$-simplex as follows, $$\triangle^2= pe_1+qe_2+re_3 \hspace{4mm} p,q,r \in \mathbb{R}$$ I want to project this triangle down to $\mathbb{R}^2$, that is so I can write ...
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1answer
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Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
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1answer
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How to efficiently determine whether or not there is a collision between two 3D triangles?

What formula can efficiently tell if two 3D triangles collide or not?
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1answer
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Problems on vector spaces

Let $E$ a $\mathbb{K}$-vector space of finite dimension $n$, $\mathcal{V}$ a subspace of $\mathcal{L}(E)$ such that $$\forall u\in\mathcal{V}\setminus \{0\},u\in\mathcal{GL}(E)$$ a) Show that ...
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Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a ...
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1answer
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For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
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Finding the 2 point coordinates for a known edge.

Say I have an edge A'B' which is a vector (5,3 9). How can I find the individial points A' and B' from A'B'. I translated the points A and B by a vector then combined them to make the edge AB. Then ...
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1answer
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Same quadratic forms on $\mathbb R^n$

Let $q$ be an inner product on $\mathbb R^n$ and $Q$ be its matrix expressed in the canonical basis of $\mathbb R^n$. Assume that the group $$SO(q)=\{A\in M_n(\mathbb R) \ | \ A^TQA=Q\}$$ of ...
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Explicit Isomorphism between Vector Spaces

Let $V$ and $W$ be two finite dimensional spaces. I want to show that I have a canonical isomorphism from the space of bilinear forms $\mathcal{B}= \left\lbrace B: V^* \times W^* \rightarrow ...