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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41 views

$\text{Ker}A=\text{span}(u) \implies A=mat_C\left( u\wedge . \right)$

i found this equality and i wonder how can i find the right term $$\dfrac{1}{2}\left(\begin{matrix}0&1&1 \\ -1&0&1\\ -1&-1&0 ...
-3
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0answers
46 views

Prove the entries in the exponential of a matrix are finite real numbers [on hold]

Let $A$ be an $n\times n$ matrix. Show that all entries of $\exp(A)$ are finite real numbers. I know that $\exp(A)=I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots$ How do I show this? Any help is much ...
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0answers
27 views

orthogonal complement and direct sum proof

I don't know how to prove this statement. Someone can help me with is. Thanks!!!
2
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1answer
89 views

Estimating rank and nullity of the composition of linear maps

Let $T\colon U\to V$, $R\colon V\to W$ be linear maps between finite dimensional spaces $U$, $V$, $W$, and let dim$(V)=n$. Prove that $\dim\, \ker(RT)\le \dim\, \ker(R)+\dim\, \ker(T)$, ...
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2answers
21 views

Dimension of a subspace of polynomials

I saw this question in a textbook. I am facing difficulty in solving it. Let $V$ be the vector space of all real polynomials and $W$ be the subspace generated by $$a x^2+b x+c$$ where $a,b,c$ take ...
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0answers
24 views

Finite field versus real ranks

Given $A\in \Bbb Z^{m\times m}$ with maximum absolute entry bounded by prime $p$. Take $q$ be prime such that $q<p$. Take $A_{(r)}$ be matrix $A$ whose entries will be remainder modulo $r$. Denote ...
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0answers
23 views

let $x,y$(non-zero) $n\times 1$ vectors .What are the eigen values of $xy^t$?

let $x,y$(non-zero) $n\times 1$ vectors .What are the eigen values of $xy^t$? Now $xy^t$ is an $n\times n$ matrix and completely arbitrary.How to find the eigen vectors?
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2answers
33 views

Proof with orthogonal matrix

I stuck at this problem: I need to prove that for linear transformation $$T:R^n\to R^n$$ defined by $$T(x)=Px$$ such that $$P^T=T^{-1}$$ for any $x,y$ $$T(x) \cdot T(y)=x\cdot y$$ and also that $T$ ...
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0answers
48 views

Clarification of definition of tensor product

I am reading "Riemannian Geometry" by Gallot. And I am confused with the following definition of tensor product: Let $E$ and $F$ are two finite dimensional vector spaces, a vector space $E\otimes ...
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0answers
23 views

GMRES and Preconditioning

I am using GMRES to approximate the solution of a system of equations $Ax=b$, I am using a preconditioner $P$ to make GMRES converge faster. My question is how do I know if the preconditioner I am ...
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1answer
25 views

Significance and physical meaning of diagonalization of linear maps and bilinear forms, eigenvalues and eigenvectors

In linear algebra, I have studied the diagonalization of a linear map and of a bilinear form; and also the concepts of eigenvalues and eigenvectors. However, the importance of diagonalizing a linear ...
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1answer
16 views

determine the point of intersection on a facet in n-dimensions

I'm trying to solve a what I think is a classic line/plane intersection problem. However, this type of problem is new to me so please excuse me if I am misusing the terminology. I have two points in ...
2
votes
1answer
57 views

determinant of matrix $X$

Please hint me. ‎How ‎can I ‎calculate ‎determinant ‎of ‎matrix ‎‎$‎X‎$‎?‎ \begin{equation*}‎ ‎\mathbf{X}=\left(‎ \begin{array}{ccc}‎ A&B&‎\cdots&B\\‎ B&A&‎\cdots& B\\‎ \vdots ...
2
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1answer
37 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
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2answers
70 views

A question about Linear Transformations from Golan's Linear Algebra book

I have found the following problem from Golan's Linear Algebra.. book. Let $\alpha,\beta:V \to W$ be two linear transformations between two vector spaces $V$ and $W$ defined over the same field ...
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0answers
23 views

Eigenvalues of Overlapping block diagonal matrices

I look for eigenvalues of general overlapping block diagonal matrices. e.g. $$\left[ \begin{matrix} 1 & 4 & 0 & 0 & 0 & 0\\ 4 & 2 & 3 & 2 & 0 & 0\\ 0 & 3 ...
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4answers
47 views

Show that $T(\mathbf x)=\mathbf 0$ has a nontrivial solution

This question in my book Let $T:\mathbb{R}^n\to \mathbb{R}^m$ be a linear transformation. Suppose $\{\mathbf{u}, \mathbf{v}\}$ is a linearly independent set, but $\{T(\mathbf u), T(\mathbf v)\}$ ...
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2answers
31 views

Relationship between the four fundamental sub spaces

I am self studying linear algebra from Gilbert strang. I can understand the dimensions of the four subspaces but I am having trouble understanding the four subspaces from the perspective of linear ...
2
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2answers
81 views

Find smallest $n \in \mathbb{N}$ s.t $A^n=I$

Let $A$ be $2 \times 2$ matrix: $$ \left( \begin{matrix} \sin\frac{\pi}{18} \\ \sin\frac{4\pi}{9} \end{matrix} \begin{matrix} -\sin\frac{4\pi}{9} \\ \sin\frac{\pi}{18} \end{matrix} \right) $$ ...
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0answers
30 views

Proof: A+B is upper triangular [on hold]

Assume A and B are nxn matrices. Prove that A+B is upper triangular.
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1answer
27 views

Is this set of functions a vector space?

I'm starting to learn linear algebra am an learning what is and what is not a vector space. I'm trying to figure out if the following set of functions is a vector space: {f : R → R | f(3) = 0} I ...
2
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1answer
29 views

How to find the inverse of the matrix over $\mathbb Z_5$

How to find the inverse of the matrix over $\mathbb Z_5$ $$ \left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right) $$
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1answer
40 views

About $ A^{-1}$ where A is 10x10 matrix

Let A be 10x10 invertible matrix with real entries s.t sum of each row is 1. Then which of follwing is true: Sum of entries of each row of inverse of A is 1. Sum if entriez of each column of inverse ...
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1answer
27 views

What is the intuition behind the reduced row echelon form of a matrix?

When we convert a matrix into reduced row echelon form , the linearly independent vectors in the pivot columns form a unit vectors in the corresponding columns ? what is really happening here if I ...
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2answers
39 views

Prove that$a^2+b^2$ is composite from the information provided.

Suppose $\alpha$,a,b are integers and $b\neq-1$. Show that if $\alpha$ satisfies the equation $x^2+ax+b+1=0$,then prove $a^2+b^2$ is composite. I am starting with this study course of polynomials and ...
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0answers
20 views

Linear Algebra - verification of my answer, basis for $ImT$

I'd like to verify this answer, because I think that the answer in my book is incorrect. I'll be very glad if someone could tell me, if the basis I found for $ImT$ is correct. Let : $T:R^3 ...
0
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1answer
21 views

Evaluation maps of functionals are linearly independent

Let $\mathcal{P}_n$ be a vector space of polynomials of degree less than or equal to $n$. I have shown that the evaluation map $Eval_x : f \in \mathcal{P}_n \mapsto f(x) \in \mathbb{R}$ is a linear ...
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1answer
47 views

A question about a linear algebra proof [on hold]

If $f(x)$ is a function with domain $R$ such that for all real $a, x$ it is $f(ax) = af(x)$ then there exists a real number $b$ such that $f(x) = bx$ for all $x.$ How to prove this statement?
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1answer
17 views

$B - A \in S^n_{++}$ and $I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++}$ equivalent?

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Now suppose that $A,B \in S^n_{++}$ are two ...
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2answers
25 views

Commuting operators

Let's consider a number of linear operators, defined on a finite dimensional complex vector space, which two by two commutes with each other. (the amount of them can be infinite). How to prove that ...
1
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1answer
16 views

Overdetermined system with parameters

$$\begin{cases} ax & +y &=2a\\ x &+by &= b \\ ax &+ (5-b^2)y &= 1 \tag{A,B,C (in order)} \end{cases}$$ Where $(x,y)$ are variables and $a,b$ are constants. What ...
1
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3answers
61 views

Proving that $(u+v)×w=u×w+v×w$

Let's $$(\overrightarrow{u}+\overrightarrow{v})\times\overrightarrow{w}=\overrightarrow{u}\times\overrightarrow{w}+\overrightarrow{v}\times\overrightarrow{w}$$ How to prove it? Update: The problem is ...
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1answer
20 views

How to prove the $i$th coordinate of $u$ in the basis equals $(u,e_i)$?

Let $e_1, e_2, e_3$ be an orthonormal basis. How to prove that for any vector $u$, $$u = (u, e_1)e_1 + (u, e_2)e_2 + (u, e_3)e_3,$$ i.e., the $i$th coordinate of $u$ in the basis equals $(u,e_i)$?
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0answers
31 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
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1answer
16 views

Diagonal matrix basis

Let $e_1,e_2,e_3$ be basis of V and $\phi$ belongs to Hom(v).Find basis for V for which $\phi$ has diagonal matrix D,so as this diagonal matrix,if for $\phi$ it is true: $\phi({a_1e_1+a_2e_2+a_3c_3}) ...
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0answers
17 views

System of equations with parameters

First class of linear algebra, and I've encountered this problem which I just can't figure out. The following system of equations has more equations than variables $(x,y)$. The parameters $a,b$ can ...
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1answer
49 views

Matrix Representation (linear algebra)

$A:X \to X$. Find the matrix representation in the basis $\mathbb{R}^2=span{(1,3), (2,5)}$ for $A(x)=(2x_2,3x_1-x_2)$ I don't know how to find matrix representation. Someone can help me and show the ...
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0answers
41 views

How advanced should mathematical statements be for undergraduate research opportunities in mathematics?

I'm applying for a few undergraduate research opportunites in mathematics this Spring, and part of the application is discussing "the most interesting mathematical result you know of." All of the ...
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2answers
27 views

Three equations, three unknowns, and one constraint

Suppose we have the following three equations: $$ r_y = \frac{r_y}{2} + \frac{r_a}{2} \\ r_a = \frac{r_y}{2} + r_m \\ r_m = \frac{r_a}{2} $$ We also have additional constraint for uniqueness: $$ r_y ...
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1answer
26 views

Lin Alg 100-Level Recursion Problem

I want to pave a $2\times n$ rectangle with $1\times 2$ blocks which come in two colours, white and grey. Let $w_n$ be the number of different ways this can be done. I determined the recursive ...
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1answer
49 views

Finding orthogonal matrix that maps one vector to another

Let $w, v \in \mathbb{R}^k$ be two known vectors such that $||w|| = ||v||$ ($|| . ||$ is the usual Euclidean norm). My questions are related with the problem of finding $Q$ orthogonal such that $v = Q ...
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5answers
114 views

$\det(I+\epsilon V)=1+\operatorname{trace}(V)\epsilon+O(\epsilon^2)$

How to show that $$\det(I+\epsilon V)=1+\operatorname{trace}(V)\epsilon+O(\epsilon^2)$$ for any $n\times n$ real matrix $V$? This is used a lot in the theory Lie groups, but I never saw a proof of ...
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0answers
34 views

Can these characterisations of finite dimensionality be proven equivalent without using a basis?

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me: Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow ...
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1answer
25 views

Kernel and image of a linear map (with parameter)

Let $T: \mathbb{R^3} \to \mathbb{R^4}$ such that $f(1,1,0) = (1,h,1,0)$ $f(0,2,0) = (1,h,1,0)$ $f(0,1,-1) = (h,2,1,1)$ I have to determine the kernel and the image of $T$ for $h \in \mathbb{R}$. ...
2
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2answers
74 views

how they deduce that $\det A=1$ just from the first coeffcient and minor

i found solution of exercice that said show that A is rotation to do that we have to compute det A=1 but they found it directly Is there any relationshipe between the first coeffcient and minor ...
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1answer
17 views

proving linear dependence proof in reverse direction

The following theorem is from my textbook: Theorem 1.7. Let $S$ be a linearly independent subset of a vector space $V$, and let $v$ be a vector in $V$ that is not in $S$. Then $S \cup \{v\}$ is ...
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1answer
22 views

Determinant of 2 transpose matrix A and B.

Can you show me why $\det(A^T B^T) = \det(A)\det(B^T) = \det(A^T)\det(B)$ ? im really having a hard time finding its properties. i dont know what to search. please help.
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1answer
17 views

Rewrite an expression in terms of basis vectors

Given any vector k $\epsilon$ $R^{3}$ consider k= $\sum_{j=1}^{3}$ $c_{j}u_{j}$ where $u_{1}$,$u_{2}$,$u_{3}$ are the orthonormal basis vectors (I don't know how to make them bold sorry about that, ...
0
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0answers
20 views

is $\mathbb{Z} + p_1\mathbb{Z} + p_2\mathbb{Z} + p_3\mathbb{Z} = \mathbb{Z} + q_1\mathbb{Z} + q_2\mathbb{Z} + q_3\mathbb{Z}$? [on hold]

My question is if $\mathbb{Z} + p_1\mathbb{Z} + p_2\mathbb{Z} + p_3\mathbb{Z} = \mathbb{Z} + q_1\mathbb{Z} + q_2\mathbb{Z} + q_3\mathbb{Z}$ as sets when $p_1, p_2, p_3, q_1, q_2, q_3$ are irrational ...
0
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1answer
24 views

Matrix transformation conserving the “positive semi-definite” aspect

Let's say I have two covariance matrices $A$ and $B$ (so they're both positive semi-definite), What kind of transformations can I apply on either one of them or both without loosing the ...