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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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
13 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
17 views

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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39 views

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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18 views

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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6answers
4k views

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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1answer
24 views

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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15 views

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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10 views

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
22 views

$\{id_V,T,T^2,…,T^d\}$ 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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2answers
16 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 $\mathbb{R}^\mathbb{R}$. I immediately thought of the Chebyshev polynomials : $T_n(x)= ...
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0answers
5 views

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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1answer
22 views

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

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$ such that $(f,f',\dots,f^{(n-1)})$ be linearly independent and $f^{(n)}=f$. Could you give me some hints ? I truly have ...
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13 views

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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12 views

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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9 views

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
46 views
+50

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
45 views

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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2answers
33 views

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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0answers
21 views

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
14 views

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
26 views

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
34 views

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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0answers
35 views

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
18 views

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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1answer
50 views
+50

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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0answers
4 views

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
25 views

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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40 views

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 ...
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3answers
24 views

Find a point $p_1$ on the line $l$ with distance d from the point $p_2$ on the same line

I have tried to find posts that are related to the question but they end up with the terms like ‘find a distance’. What I want is not to find the distance: I already have the distance, I want ...
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2answers
39 views

The Number of Increasing Vectors $(x_1,…,x_k)$ Satisfying $1 \leq x_i \leq n$ and $x_1 < x_2 <…<x_k$

I want to find the number of increasing vectors $(x_1,...,x_k)$ satisfying $1 \leq x_i \leq n$ and $x_1 < x_2 <...<x_k$. Examples of vectors satisfying these conditions Let $n =5$ and ...
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2answers
206 views

A determinant problem

If $f(n)=\alpha^n+\beta^n$ and $$A=\left| \begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array} \right|$$ ...
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3answers
41 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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2answers
35 views

I am not getting the concept actually of dimension of intersection of subspaces

Let $$W_1=\{(0,x_2,x_3,x_4,x_5)|\forall x_i\in \Bbb R, i=2,3,4,5 \} $$ $$W_2=\{(x_1,0,x_3,x_4,x_5)|\forall x_i \in \Bbb R, i=1,3,4,5\}$$ be subspaces of $\mathbb{R}^5$ then what is $\dim(W_1 \cap ...
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1answer
48 views

Analogy between linear basis and prime factoring

I recall learning that we can define linear systems such that any vector in the system can be represented as a weighted sum of basis vectors, as long as we have 'suitable' definitions for addition and ...
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1answer
35 views

Jacobian and PDE

I am wondering how to compute the Jacobian in order to know if a given PDE satisfying an initial condition has a unique solution or not. If I consider the PDE, $u_x=1$, satisfying the initial ...
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1answer
20 views

Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...
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1answer
244 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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2answers
17 views

Characteristic polynomial factor over the real numbers

Ve=the set of symmetric 2x2 matrices I'm trying to show that any element of Ve has a characteristic polynomial that factors over the real numbers and has two distinct eigenvalues unless the matrix ...
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1answer
15 views

Unique linear combination and basis

Let $S \in \mathbb{R}^3$ be the following set of vectors $$ v_1=\begin{pmatrix} 1\\ 0\\ 1 \end{pmatrix} , v_2 =\begin{pmatrix} 0\\ 1\\ 1 \end{pmatrix}, v_3 = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} ...
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5answers
137 views

Symmetric and Skew Symmetric Matrices

$V_0=$ the set of $2\times2$ skew symmetric matrices I know that any element of $V_0$ has a characteristic polynomial that will not factor over the real numbers, and therefore has no eigenvectors. ...
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9answers
279 views
+100

Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
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1answer
21 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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0answers
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$|y-x|\lt \epsilon$ and $|z-x|\lt \epsilon$

I'm trying to solve this question: Is it not obvious we can arbitrarily approximate the points $y$ and $z$ to $x$? but how to formalize this? Any help? hints to begin with? Thanks Notation ...
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1answer
34 views

Inverse of a diagonal matrix plus a constant

I am looking for an efficient solution for inverting a matrix in the following form: $D+aP$ where D is a (full-rank) diagonal matrix, a is a constant, and P is a one matrix. This question Inverse of ...
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2answers
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Questions on symmetric matrices [on hold]

Describe a basis for the vector space of symmetric n x n matrices. What is the dimension of this space?
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3answers
33 views

dimension of a symmetric matrix with trace zero

what will be the dimension of symmetric matrix of order $n\times n(n\geq2)$ with real entries and trace is equal to zero? The answer is given as :$\frac{n^{2}+n}{2}-1$ can anyone explain how will ...
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2answers
186 views

Is Im(T) + Ker(T) the same as Im(T) union Ker(T)

If i know Im(T) and Ker(T), is Im(T) + Ker(T) the union of the two vector space? If not, how do i find the addition of the two vector space. It is best if examples can be given. Thanks.
2
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1answer
80 views

Evaluate this covariance matrix.

Edit: I have added an approach provided by @GiannisChantas. It would be great (and much appreciated) if someone could check if this approach is correct! I have also added a secondary question for ...
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3answers
640 views

Difference between span and basis

What is the difference between the span of the image of a matrix and the basis for the span of the image of a matrix? Are these the same thing?