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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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 $\mathbb Z_5$ $$ \left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right) $$
2
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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
28 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 ...
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1answer
22 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
18 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
26 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 ...
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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 ...
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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
42 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 ...
2
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1answer
51 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 $h \in \mathbb{R}$. ...
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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, ...
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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 ...
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0answers
20 views

Candidate inner product [on hold]

Consider the space $P_2(\mathbb{R})$ of (real) polynomials of degree two or less with the (candidate) inner product: $$ \langle f,g \rangle = af(0)g(0)+ \int_0^1 f'(t)g'(t) \text{d}t $$ where $f'$ ...
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1answer
24 views

Is LU decomposition of matrices efficient for today's standards?

This is in the spirit of a previous question of mine about the efficiency of the QR algorithm. The reason for asking is that I want to motivate some students, and I'm also curious. I do understand ...
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1answer
22 views

Equalities of rank inequalities

Why do I ask this question? Since I got into trouble with the problem below: If $A,B$ are $n\times n$ matrices which satisfy $A^{2013}=0, AB=BA, B\neq O$, then $$rank(AB)\le rank(B)-1$$ I don't ...
2
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1answer
41 views

Is a subset a vector space if it is just one vector?

I'm beginning to learn linear algebra and wanted to know if a subset of a vector space is a vector space if there is only one vector. For example, $$V = \{(x, y, z) \in \mathbb{R}^3| x^2 + y^2 + z^2 ...
2
votes
1answer
44 views

proof on similarity of matrices

Could you please help me with the following problem? Let $A$ be an $n$$\times$$n$ complex matrix. Prove that $A$ is similar to $B$, which is an $n$ $\times$ $n$ real matrix, if and only if $A$ is ...
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1answer
28 views

Is the inverse of a causal function causal?

I am wondering if the inverse of a causal function is causal. I'll illustrate what I mean with an example: Assume $f$ is a bijection of $\mathbb R^2$ in $\mathbb R^2$. I assume $f$ is causal in the ...
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1answer
29 views

If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
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2answers
29 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
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1answer
40 views

Correct proof of $\det(X+iY) \det(X-iY)>0$?

Can someone please look over my proof below as to why $\det(X+iY) \det(X-iY)>0$ for real matrices $X,Y$, such that $\det(X+iY)$, $ \det(X-iY)$ not both the zero, and tell me if it's correct ? My ...
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0answers
13 views

Find the basis of the matrix

Find the basis.I got all must be 0? $$\phi = \left[\begin{array}{cccc} 4 & 1 & 1 & 1\\ 1 & 4 & -1 & -1 \\ 1 & -1 & 4 & -1 \\ 1& -1 & -1 & 4 ...
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2answers
15 views

Basis of matrix

What is the basis of ker of $$\phi = \left[\begin{array}{cccc} 0 & -4 & -4 & 0\\ 0 & 0 & -8 & -12 \\ 0 & 0 & 0 & -12 \\ 0& 0 & 0 & 0 \end{array}\right] ...
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0answers
28 views

Diagonal matrix zero elements

In orthogonal basis,Suppose we search on a given matrix the diagonal matrix and writing on every cell of the diagonal $\lambda $.When we find the determinant we find a value for $\lambda$ and it ...
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1answer
30 views

Subspace and orthogonal complement span whole space even if the form is degenerate?

Let $V$ be a finite dimensional vector space over a field $\mathbb{R}$. Let $B\colon V\times V\rightarrow \mathbb{R}$ be a bilinear form. Assume that $B$ is degenerate. Q.1 Does $V$ has orthogonal ...
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1answer
49 views

Is this a subspace? $X = \{x \in \mathbb R^2: x \cdot y = 0\}$

I am completely lost in this question and hoping someone could explain this question through for me. Let $y=[1,2]$ $X = \{x \in \mathbb R^2: x \cdot y = 0\}$ Is $X$ a subspace? So I get the ...
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1answer
20 views

How to find the transition matrix from basis $E$ to $E'$

Suppose there is a linear transformation $T$ on $\mathbb R^n$. And $$E=[\epsilon_1,\epsilon_2...\epsilon_n]$$and $$E'=[\epsilon'_1,\epsilon'_2,...\epsilon'_n]$$ are two different basis of $\mathbb ...
3
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1answer
63 views

Invariance of the Fredholm index under finite-dimensional perturbations

Let's call a linear map $f : V \to W$ between vector spaces over some field Fredholm if $\ker(f)$ and $\mathrm{coker}(f)$ are finite-dimensional. (Equivalently, it represents an isomorphism in the ...
5
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1answer
39 views

$|\det (A+B)|\ge |\det B|$ for all $B$ such that $AB=BA$ iff $A^2=O$

Let $A\in M_2(\mathbb{C})$. $Z(A)$ is the set of all $B\in M_2(\mathbb{C})$ such that $AB=BA$. Prove that $|\det(A+B)|\ge |\det B|$ for all $B\in Z(A)$ if and only if $A^2=O$. If $A^2=O$ and ...
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2answers
57 views

Prove linear independence of a set $\{\mathbf{x}-\mathbf{x_1},\ldots,\mathbf{x}-\mathbf{x_n}\}$

Let $V$ be a vector space and suppose that $\{\mathbf{x_1},\ldots,\mathbf{x_n\}}$ is a linearly independent subset of $V$. If $\mathbf{x} = \sum_{i=1}^n c_i\mathbf{x_i}$ where each $c_i \in ...
2
votes
1answer
53 views

Example of a non singular square matrix such that $A+A^{-1} = 0$

Is there any example of a non singular square matrix $A$ such that $A+A^{-1} = 0$? Are they any specific type of matrices or can these be found under any category of matrices (such as symmetric, ...
2
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1answer
35 views

Representation theory of $\mathbb{Z}_k$ and complex roots of unity

Is there a natural relationship between the (characters?) of irreducible representations of $\mathbb{Z}_k$ and the $k$ complex-roots of unity? Can they be like thought of as characters of its ...