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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Rayleigh quotient ($|r(q)-\lambda|=O(||q-x||_2^2)$ ?)

how to show $|r(q)-\lambda|=O(||q-x||_2^2)$ $r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, A is a Hermitian matrix. x is the unit eigenvector corresponding to $\lambda$. and q is a unit vector. ...
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Approximate matrix by a rank 2 matrix using singular values

I only understand the singular value decomposition process. Do I apply it to this matrix? \begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix} What is the idea ...
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Showing that the magnitude of the difference of two vectors is larger than the difference of it's vector magnitudes

Long title. I have to prove (the problem itself suggests using Pythagorean theorem) the following inequality: $$\|u\|-\|v\| \le \|u-v\| $$ Vector magnitudes... How do you prove this in an ...
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The geometric meaning of a line plus a vector

Lets say we have $$ E = \{k(1,2,3)' + (2,9,-1)'\} \;\mathrm{with}\; k \in \mathbb{R} $$ we know that $k(1,2,3)$ spans a line in three dimensions, but what does the shape of $E$ look like. I think it ...
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composition of transformations P_1 --> P_2

let $T_1 : P_1 \to P_2$ be a linear transformation defined by $T_1(p(x))=xp(x)$ and $T_2:P_2 \to P_2$ be a linear operator defined by $T_2(p(x))=p(2x+1)$ $B=\{1,x\}, B'=\{1,x,x^2\}$ a) Find $[T_2 ...
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36 views

Can anyone explain how we use the linear extension theorem?

I am having problems using the linear extension theorem. For example: Let V be finite-dimensional, and let W ⊂ V be a proper subspace of V . Fix a vector v0 ∈ V such that v0 is not in W. Show that ...
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33 views

Every Solution set of Homogeneous system is a linear combination of fundamental solutions

Prove: Every Solution set of Homogeneous system is a linear combination of fundamental solution. can I say that the fundamental solution is a trivial basis therefore is spans the Null space? can I ...
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Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$.

Let the subspace $V_o \subset V$ be the image of $V$ under $A$. Let $k = \dim (V_o) \lt n$ and suppose that for some $\lambda \in \mathbb{R}$, $A^2 = \lambda A$. Then which of the following are true? ...
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What is $\Bbb{R}^n$?

I earlier asked this question The basis of a matrix representation. I now have a another question related to the same topic. The vector space $\Bbb{R}^n$ I have seen defined as all $n$-tuples of real ...
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Finding a base field for diagonalized linear transformation and justifying

So I encountered this long question that asks you to find bases and such, I searched up Find the eigenvalues for the linear transformation and base associated to each eigenvalue. If possible find ...
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T is a diagonalizable linear operator on a finite dimensional vector space V. Then every linear operator which commutes With T is a polynomial of T

I'm trying to answer this question True or false? T is a diagonalizable linear operator on a finite dimensional vector space V. Then every linear operator which commutes With T is a polynomial of T. ...
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How can I rewrite this expression?

$A=\begin{pmatrix}3&-1\\-1&1\end{pmatrix}$; $U_\phi=\begin{pmatrix}\cos\phi&-\sin\phi\\\sin\phi&\cos\phi\end{pmatrix}$; ...
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The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
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Solution to a system of linear equations with an unknown matrix product

Consider the system of equations $$ Xy=Ab $$ where $X$ and $A$ are $m \times m$ invertible matrices and $y$ and $b$ are $m \times n$ matrices. The matrices $X$ and $y$ are unknown and the matrices $A$ ...
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Number of parameters on the general solution of a differential equation

I have the following differential equation : $c_1$.x'' + $c_2 $.x = 0 . Being $w=\sqrt{ c_1/c_2 }$ I was told that the general solution can be either $x(t) = A.cos(wt + \phi_1 )$ ...
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Why is the projection of a vector V onto a span W, independent of the orthogonal basis of W.

Very straightforward question. I have read time and again in my book that it is independent but I don't understand why? Wouldn't changing the basis mean changing the length of the projection?
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Maximize the difference of two linear expressions

Given two $1\times N$ complex vectors h and g. I want to find a $N\times 1$ complex vector w(normalized to unit norm $ \Vert w \Vert^2=1$), which maximizes the following expression: $$w_0=\arg\max_w ...
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102 views

Interlacing of eigenvalues for Hermitian matrices

This is a problem from Matrix Analysis by Horn and Johnson. Let $A \in M_n$ be Hermitian, let $a_k$$=$det$A$[{$1$, $\dots$,$k$}] be the leading principal minor of $A$ of size $k$, $k = 1, \dots, n$, ...
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Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.6, Problem 11

Let $X$ be the vector space of all complex $n \times n$ matrices and define $T \colon X \to X$ by $Tx \colon= bx$, where $b \in X$ is fixed and $bx$ denotes the usual product of matrices. I know that ...
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Question on the definition of vector spaces.

My question is perhaps useless, but I want to shed some clarity on this matter. I'm bothered by people that say a vector space is a "bunch of vectors". Or that a vector space "consists of ...
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56 views

Linear maps that are matrices

If I have the linear map $A:\Bbb{R}^3\rightarrow \Bbb{R}^3$ where $A$ is a matrix. Is the matrix $A$ (along with the vectors it operates on) in a basis or not? I think it is not, since the vectors it ...
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Solution set of Homogeneous systems (Wikipedia Error?)

In the definition of the solution set of Homogeneous systems in Wikipedia it is written: Every homogeneous system has at least one solution, known as the zero solution (or trivial solution), which ...
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Vector Spaces and Groups

I've just completed a course in linear algebra. I'm a physics undergraduate and I don't plan on taking an abstract algebra course. That said, I've been reading a little bit about it. As I understand ...
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Given similar matrices $A$ and $B$, how to find $M$ such that $B=M^{-1}AM$?

I am trying to teach myself linear algebra using Strang's Introduction to Linear Algebra. I would like to know what the most (or more) efficient way to solve this problem is by hand. The question: ...
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51 views

Linear Algebra question about orthogonal projection (Upper Linear Algebra)

Definition: The orthogonal projection of $V$ onto $U$, $P_U$, is defined by $P_U(v) = u$, where $v = u + u'$ for $u ∈ U$ and $u' ∈ U^{\perp}$. Furthermore, if $(e_1, \ldots, e_m)$ is an orthonormal ...
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Find the orthonormal vectors $q_{1}, q_{2}, q_{3}$ such that $q_{1}, q_{2}$ span the column space of $A$?

We have given the matrix $$ A= \begin{pmatrix} 1 &1 \\ 2& -1 \\ -2 & 4 \end{pmatrix}$$ First the question asks find the orthonormal vectors $q_{1}, q_{2}, q_{3}$ such that $q_{1}, ...
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Find the determinant of a matrix definition [duplicate]

Let $A$ be a matrix that is defined like this: $$A_{ij}=\begin{cases} \alpha, & \text{if i=j} \\ \beta , & \text{if i $\ne$ j} \end{cases} $$ So I realized this matrix looks somehow like ...
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Dim(V) >= Nullity(T)

If $T:V \to W$ is a linear transformation the $nullity(T) \le \dim(V)$ I'd like to say its true but I need a proof to show it. Just not sure where to start. I think whats throwing me off is the ...
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Optimization Problem - Lowest Total Price from Multiple Suppliers

I believe this is a linear algebra problem, but if not please let me know: Say you have 4 suppliers. You want to order 4 different items. The 4 suppliers each have a different price for each item and ...
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284 views

Easy way to calculate the determinant of a big matrix?

Given this matrix: \begin{matrix} 2 & 3 & 0 & 9 & 0 & 1 & 0 & 1 & 1 & 2 & 1 \\ 1 & 1 & 0 & 3 & 0 & 0 & 0 & 9 & 2 & 3 & ...
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How many distinct integer solutions does the inequality $|x_{1}|+|x_{2}|+…+|x_{n}| \leq t$ have?

How many distinct integer solutions does the inequality $|x_{1}|+|x_{2}|+...+|x_{n}| \leq t$ have? We know that: $x_{i} \in Z,\ \forall i \ 0\leq i \leq n \ and \ t\geq0.\ $ I know that if we ...
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1answer
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What is the dimension of $A-B$, where $B$ is a subspace of $A$?

My question is really simple, what is the dimension of $A-B$, where $B$ is a subspace of $A$? this space is well-defined? I found this space in this paper on page 440: Following my calculations in ...
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What is the linear space of Eigenvectors associated with a certain Eigenvalue?

The following matrix $A$ has $\lambda=2$ and $\lambda=8$ as its eigenvalues $$ A = \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{bmatrix}$$ let $P$ be the ...
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Algebraic characterization of a union of two affine subspaces

Is it a simple algebraic characterization of affine hull of the union of an affine set $A$ in a linear space and a point $x$ not lying in this hyperspace? I thought that it is $$\{(1-t)a+tx: t\in ...
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Eigenvalues of a nilpotent matrix can only be $0$ [duplicate]

Prove that the eigenvalues for a square Nilpotent matrix A can only be $0$. Definition of nilpotent A $^n$=$0$ n is a positive whole integer
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Under what assumptions is it correct to say “a matrix is diagonalizable if and only if its eigenvalues are real”?

A $2\times 2$ matrix is diagonalizable if and only if its eigenvalues are real. Which statement is most correct: The proposition is true only if the eigenvalues are all greater than zero. The ...
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Finding polynomial generators in a subspace

$S$ is a subspace $S= \{p\in P_3|~\text{$i\in\Bbb C$ is root of $p$}\}$. So the question at hand is how do you find the system of generators for the subspace knowing that $x$ is $p$'s divisor? ...
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Algorithm for the Hill cipher (finding the inverse of the determinant of a $2 \times 2$ matrix modulo $26$)

I have a good understanding of how to do the Hill cipher on paper but putting it into program form is somewhat of a problem. Finding the the determinant is the thing I'm having problem with. On ...
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42 views

Converting nth order ODE with RHS into system of 1st order ODEs

I looked at these two questions, but they weren't directly relevant to my specific question: How to reduce higher order linear ODE to a system of first order ODE? Express differential equations as ...
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Under what condition on a set $S$ does $f$ exist such that $f \cdot S_k$ has the same value for all $S_k\in S$?

this is my first time using this site so I apologize if I'm unclear or using poor convention. I'm working on a problem with wireless power transfer, which long story short involves a set of transfer ...
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Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
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Prove there exists a sequence of bounded numbers $(v_1,v_2,..)$ such that $\sum_{n=1}^\infty a_nv_n = f(x)$?

Let $(X,\|\cdot\|)=(l^1,\|\cdot\|_1)$ and let $f$ be a bounded linear functional. Prove there exists a bounded sequence $(v_1,v_2,...)$ of real numbers such that: ...
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563 views

Which of the following subsets of P2 are subspaces of P2?

{p in P2: p(0) > p(1)} {p in P2: p(3) = p(4)} {p in P2: p'(3) = 4p(7)} I understand that generally we need to check to make sure that these are closed under addition and scalar multiplication, and ...
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linear algebra, space vectors

I am having difficulties in proving whether a particular E is space for example IR, so I would like if you could do this exercise. Determine whether IR ^ 2, with the operations described, is a real ...
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Find the orthogonal projection P on L.

Let $L=<3e_3+2e_1,e_5>$ and $x=(2,1,3,2,-6,8,2,1,0,0,0,...)$. Find $\|Px\|$ where P is the orthogonal projection on L (Do not forget that the formula for Px orthogonal vectors need length 1) I ...
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Find an orthogonal matrix that achieves a given vectorial transformation

Given a vector $\vec a\in\mathbb R^n$ and another $\alpha=(\|\vec a\|,0,\dots,0)$, how could I define an orthogonal matrix $M$ such that $M\vec a=\alpha$ and $M^{-1}=M^t$? For $\mathbb R^2$ I tried to ...
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How prove this rank identity $r(A)=r(B)$

let $A_{n\to n},B_{n\to n}$ matrix,and such $$A^2=20142014A,B^2=20142014B,$$ and $20142014I-A-B$ is invertible, show that $$\rm{rank{A}}=\rm{rank{B}}$$ we know $$A^2-20142014A=0$$ then ...
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How to find the characteristic polynomial of this transformation?

Let V be a finite-dimensional inner product space, and let W ⊂ V be a subspace. Let T : V → V be the linear transformation “orthogonal projection onto W”: T(x) = ProjW x. Show that T is ...
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40 views

$p(x)$ divides the minimal polynomial iff $\exists v\ne 0: p(T)(v)=0$

Let $V$, a finite dimensional space. Let $T:V\to V$ a linear transformation. Show that $p(x)$, an irreducible polynomial divides $m_T$ (The minimal polynomial of $T$) iff there is a $V\ni v \ne 0$ ...
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28 views

Give an example of a linear subspace $L$ of $H=l^2$ such that there exists no $y\in L$ such that $\|x-y\|=$dist($L,x$)?

I know that a subspace that is closed and convex must have a unique y in L such that it is true and that if it is closed presumably then you can have many y. So I am looking for an L which is open in ...