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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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 ...
-2
votes
3answers
38 views

what is the smallest number $n\in \mathbb N$ such that $A^n=I$? [on hold]

Let $A$ be a $2\times 2$ matrix consisting of $$A = \begin{pmatrix}\sin(\pi/18)&-\sin(4\pi/9)\\ \sin(4\pi/9)& \sin (\pi/18)\end{pmatrix}$$ what is the smallest number $n\in \mathbb N$ such ...
-2
votes
0answers
45 views

$A$ be a $10\times 10$ matrix over $\mathbb R$ such that sum of each row is $1$. [on hold]

Let $A$ be an invertible $10\times10$ matrix over $\mathbb R$ such that sum of each row is $1.$ Then which option is correct? A. The sum of the entries of each row of the inverse of $A$ is ...
0
votes
1answer
18 views

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 ...
0
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1answer
26 views

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. ...
2
votes
2answers
174 views

How can I rewrite this expression?

$A=\begin{pmatrix}3&-1\\-1&1\end{pmatrix}$; $U_\phi=\begin{pmatrix}\cos\phi&-\sin\phi\\\sin\phi&\cos\phi\end{pmatrix}$; ...
3
votes
2answers
63 views

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 ...
0
votes
2answers
47 views

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 $A$ ...
0
votes
3answers
19 views

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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2answers
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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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2answers
35 views

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 ...
2
votes
1answer
48 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 $k$, $k = 1, \dots, n$, ...
3
votes
2answers
67 views

Problem 11 Section 2.6 in Erwine Kryszeg's Introductory Functional Analysis With Applications

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 ...
1
vote
3answers
61 views

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 ...
0
votes
1answer
34 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 ...
0
votes
1answer
25 views

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 ...
3
votes
1answer
48 views

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 ...
0
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0answers
22 views

Banach Algebra spectral theory [on hold]

Let $(\Omega, \mu)$ be a measure space. Show that the linear span of the idempotents is dense in $L_\infty(\Omega, \mu)$.
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0answers
33 views

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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votes
0answers
34 views

I need help with this homework problem, please help with the solution and ste by step answer. Thanks. [on hold]

Show the that indefinite integral of the delta function is the Heaviside function of f(x)= {0 if x<0 {1 if x>0 *greater than or equal to zero. This suggest that the derivative of f(x) is ...
0
votes
1answer
50 views

Quality of approximation [on hold]

Basically i have written a code in matlab that is used to solve lower triangular system with forward elimination Ax=B. However, the forward elimination script will not give the exact vector but a ...
0
votes
1answer
27 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 ...
4
votes
5answers
122 views

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 ...
0
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0answers
32 views

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 ...
0
votes
1answer
36 views

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 ...
2
votes
2answers
156 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 & ...
2
votes
1answer
67 views

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 ...
2
votes
1answer
35 views

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 ...
1
vote
1answer
42 views

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 ...
0
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0answers
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Definition of well-defined for special case

I have a question about what well-defined means in a certain case. For an operator from $X$ to its dual $X^{*}$, say $A:X \rightarrow X^{*}$,why does the definition of $A$ being "well−defined" seem ...
0
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1answer
25 views

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 ...
-1
votes
2answers
58 views

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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2answers
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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 ...
0
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0answers
17 views

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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0answers
17 views

Distance between point and translated subspace

Given $(m - 1)$-dimensional subspace of $n$-dimensional space ($m \leq n$), that is defined by a set of $m$ its (linearly independent) points. How to compute the distance between separate point $p_0$ ...
0
votes
1answer
17 views

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 ...
0
votes
1answer
7 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 ...
0
votes
1answer
25 views

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 ...
0
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0answers
12 views

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 ...
0
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0answers
26 views

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: ...
0
votes
1answer
11 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 ...
0
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0answers
12 views

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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0answers
31 views

linear algebra, space vetors

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 ...
0
votes
0answers
13 views

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 ...
0
votes
2answers
19 views

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 ...
1
vote
2answers
56 views

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 ...
0
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1answer
15 views

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 ...
0
votes
1answer
27 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 $V\ni v \ne 0$ ...
0
votes
2answers
20 views

Linear Transformation of standard matrix [closed]

Please help me for solving following problem a) find the standard matrix $A$ for the linear transformation $T$, b) Use $A$ to find the image of the vector $v$. $T$ is the counterclockwise rotation ...
0
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1answer
21 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 ...