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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number cubic polynomials possible

Let $p(x)$ be a cubic polynomial with integral coefficients , such that $p(a)=b$, $p(b)=c$, $p(c)=a$ for $a,b,c$ being distinct integers . find number of such possible polynomials.
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31 views

Linear map over a vector space of polynomials

Let $F$ be a field and Let $F_{n+1} [X]$ (odd notation, in my opinion) be the vector space of polynomials of degree less than or equal to $n$ over $F$. Define $t: F_{n+1}[X] \to F_{n+1}[X]$ by ...
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2answers
66 views

Linear operators

$\text{Hi, I am working on a assignment and I came to the solution }\\ \text{but it is not correct according to the book.}\\ \text{Can someone, please, take a look. I would really like to know what am ...
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1answer
20 views

Algebraic expression

Let us put $$c = (a + b)/2$$ Is it possible to express $$d = ab = f(c)$$ as the function of c? We can write $$ab = (a + b)^2 - (a^2 + b^2 +ab)$$ $$2ab = (a+b)^2 - (a^2 + b^2)= 4c^2 -(a^2 + ...
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1answer
28 views

A question about the subspace $U_0$ of the dual space of $V$ for a vector subspace $U\subset V$

For a vector subspace $U\subset V$ of a vector space $V$ set $$ U_0= \{f\in V^* | \forall v\in U\colon f(v)=0 \}. $$ Here $V^*$ is the dual space of $V$. I want to show that if $v\in V$ is a vector ...
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3answers
335 views

Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant,…but they are not similar.

Question: Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant, trace and characteristic polynomial, but they are not similar to each other. My thought: I come up with two ...
3
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1answer
148 views

Why do we define addition of matrices only when they have the same size

What happens if we define $$ \begin{pmatrix} 1 & 2 \\ 1 & 2 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 ...
3
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1answer
76 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
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2answers
71 views

reduction of a skew-symmetric matrix

Birkoff and MacLane state that any real symmetric matrix $A$ has the form $ A = P^{-1}BP $ where $ B^2 $ is diagonal and they ask for a proof as an exercise. It seems to me that if $A$ is ...
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1answer
84 views

Peculiar Matrix

I came up with this idea recently and I want to go deeper in this, but it has been difficult to me. Hope someone can help me on this. Suppose I have a matrix of order $(n^2-1)\times (n^2-1)$ with ...
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1answer
34 views

Finding the value of a linear combination out of a system of equations

Let $A \mathbf x=\mathbf b$ be a system of linear equations with $e$ equations and $n$ unknowns $x_1,\cdots,x_n$, s.t. $e<n$. Since there are fewer equations than unknowns, we cannot find the ...
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99 views

Linear independence of finite field elements and subfields

Let $q$ be a prime power and $n=lm$ an integer with $l,m>1$. We know that the finite field $GF(q^n)$ is a $n$-th dimensional vector space over $GF(q)$, and it is also a $l$-th dimensional vector ...
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0answers
36 views

What are the Routh Hurwtiz Criteria for 3$\times$3 Matrices?

The Criteria I know (for dynamical systems) is... The eigenvalues of a matrix are guaranteed to be negative if Tr($J$)<0 and det($J$)>0, where $J$ is the Jacobian of some 2 dimensional dynamical ...
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1answer
66 views

Distance from a point to a plane [duplicate]

How do you find the distance from a point to the plane when the point and the basis for the plane are given?
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0answers
33 views

Linear Independence of Powers of “roots vector” [duplicate]

Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$. Is the following matrix always ...
4
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1answer
88 views

An Algebraic Version of vector spaces

Consider the following set of real numbers $\mathcal{X}=\{1,2,3,\sqrt{2}+1,\pi+\sqrt{2}\}$. Lets consider the set of all linear combinations with integer coefficients of these numbers which I will ...
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2answers
88 views

isomorphisms of two vector spaces and their duals

If two vector spaces are linearly isomorphic, then so are their duals. Is the converse true? Thanks for your mention.
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1answer
58 views

Problem determining eigenvalues of a Hermitian matrix

Suppose that you've got an $n \times n$ irreducible matrix $A$ with strictly positive real entries and eigenvalues $\lambda_i$, $i=1,...,m$, arranged so that $|\lambda_1| > \cdots > ...
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1answer
40 views

How to find a matrix of a linear transformation?

Let $V = \text{span}\{(1, 1, 1),(-1, 1, 2)\}$, and let $T:\Bbb R^3 \mapsto\Bbb R^3$ be the linear transformation given by the orthogonal projection onto $V$. What is the standard matrix of $T$? Please ...
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2answers
48 views

Linear maps, matrices, nullity and rank.

I am currently trying to solve this question in my first year linear algebra course: I understand that the assoc. matrix is the coefficients, eg for (a) [[1, -1];[5, 0]], but I'm not sure how to ...
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1answer
64 views

Linear Algebra True/Flase

Are these two statements true or false, if brief justification/counterexample could be given it would be appreciated. $(1)$ $I_{V}$ is an identity operator on vector space $V$, $\dim V=n$ and $A$ is a ...
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1answer
116 views

Rank of matrix a submatrix $B$ from $A$

Question: A submatrix $B$ consisting of "s" rows of $A$ is selected from an n-square matrix $A$ of rank $r_{A}$. prove that the rank of $B$ is equal to or greater than $r_{A}+s-n$. My thoughts: I ...
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2answers
53 views

Real Life Rounding Phenomena When Solving for Variables

I have a question that I've been thinking a long time about without being able to come up with an answer and would appreciate some help: I am attempting to subtract two distinct fees from a total ...
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0answers
26 views

Decomposing isometries of $O(\mathbb{R}^3,\phi)$

This is my problem: Let be $\phi$ a nondegenerate inner product on $\mathbb{R}^3$ represented by the symmetric matrix $$A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ...
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2answers
47 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
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1answer
79 views

a question on race, time, distance

P and Q are two points on a 1km long circular track. The distance PQ along the track is 200m. Rohan started running from P and sohan started simultaneously from Q in same direction.Both reached P ...
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1answer
49 views

condition for having a positive solution to a linear equation.

Let $Y$ be a member of $\mathbb{R}^m$. I need a necessary and sufficient condition on a $n\times m$ binary matrix $A$ for having a solution to the linear equation: $$AX=Y$$ Such that $X_i\geq 0$, ...
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1answer
40 views

Is there a pseudo inverse $X$ such that $ABX=A$?

Question The title pretty much sums it up. I need to find a matrix $X$ such that: $A B X = A$, with $A\in R^{n\times n}$, $\text{rank}(A)=n$, $B\in \mathbb{R}^{n\times m}$ given. The matrix $X$ ...
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1answer
65 views

Find the basis for the null space of the given matrix and give nullity(A)?

Find the basis for the null space of the given matrix and give nullity(A)? $A=\left[\begin{array}{rrrrrr} 1 & 1 & 2 & 1 \\ 0 & 0 & 1 & -3 \end{array}\right] $ By the way, ...
3
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1answer
233 views

Name of an identity for traceless matrices in $\mathbb{R}^3$?

While working on a more compact presentation of a derivation in the context of incompressible fluid flow we tried to simplify things by introducing intermediate steps instead of writing out lengthy ...
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2answers
47 views

Number of solutions for $xy<d$, where $x,y>1$

Is there a way we can determine number of solutions for equation $$xy < d$$ where $d$ is a constant and $x$ and $y$ are positive integers greater than $1$? I am not interested in actual ...
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1answer
81 views

pseudo-inverse by SVD decomposition has not accurate results?

The goal is finding $\frac{{\partial f}}{{\partial {\bf{A}}}} = 0$ where $ f\left( {{\bf{A}},{\boldsymbol{\alpha }}} \right) = {\left( {{{\bf{p}}^{\bf{T}}}{{\bf{A}}^{\bf{T}}}{\boldsymbol{\alpha }} ...
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1answer
34 views

what is the difference in finding the basis of a subset and a basis of a null space?

I just need some explanation to what the difference between a subspace and a null space is, I think that would help me understand. Thanks!
2
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1answer
143 views

Surjectivity of an integration map

N.B.: Thanks to studiosus answer I realised I should ask for more conditions or otherwise the answer is straightforwardly wrong. I rechecked my problem and added new assumptions that I boldface. ...
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0answers
110 views

Extension field of F2 , expressing roots and primitive elements in that field

Let $\Phi$ be an extension field of $\Bbb{F}_2$ of extension degree s >1. Let $a(x)$ be a non-zero polynomial with the coefficients in $\Bbb{F}_2$. (a) Show that if $\beta$ is a root of the ...
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1answer
42 views

Linear Algebra Basis Help?

How would I write a basis for this matrix? $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end {array} \right)$ I want it in the form of $x(t)= t(?,?,?)$ What ...
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2answers
32 views

radical expression foil which confuses me

So straight to example $\sqrt{x} = 3-\sqrt{x}$ When squared gives us $x = (3-\sqrt{x})(3-\sqrt{x})$ Now foil $x = 9 - 3\sqrt{x} - 3\sqrt{x} + ?$ I don't understand the l part of the foil what ...
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2answers
57 views

When is the matrix invertable

Find the value of $k$ for which the matrix is invertable. $$ \left( \begin{matrix} 2&0&4&6 \\ k&4&3&2 \\ 3&0&1&4 \\ 2&3&0&3 \end{matrix} \right)$$ I ...
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1answer
20 views

Showing equivalence of the positivity condition of inner products

I have no idea how to prove this, first off, because I don't think I understand the question. Isn't the second case not true for v = 0. Show that for real vectors spaces $V$ with $V$ $\not= {0}$, ...
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1answer
103 views

How to prove that the rank and the nullity of similarity invariants are the same?

Given matrix $A$ and $P^{-1}AP$ how do prove that the $\mathrm{rank}(A)$ and $\mathrm{rank}\left(P^{-1}AP\right)$ are the same? Also, how do you prove that the $\mathrm{nullity}(A)$ and ...
6
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1answer
59 views

$W$ is a subspace of $\mathbb{R}^n$ and $K$ is a compact subset of $V$ with $W \cap K = \emptyset$.

Suppose $W$ is a subspace of $\mathbb{R}^n$ and $K$ is a compact subset of $V$ with $W \cap K = \emptyset$. Show that there exists a vector $v \in V$ such that $\langle v,w \rangle = 0$ for all $w\in ...
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1answer
30 views

$A \oplus C = B \oplus C$ but $A\neq B$

Let $V$ be a vector space, with subspaces $A, B,C$ such that $A\oplus C = B\oplus C = V$. Prove or give a counterexample disproving that $A=B$. I am trying to find a counterexample.
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1answer
67 views

How can i show that $\det(A)=\det(A^\intercal)$?

For all $n$-dimension matrix, how can i show that $\det(A)=\det(A^\intercal)$ by using induction? ($A^\intercal$ is a transpose of $A$)
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2answers
57 views

What are elements of a field called

From Linear Algebra by Serge Lang we have "Let K be a field. Elements of K will also be called numbers (without specification) if the reference to K is made clear by the context, or they will be ...
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0answers
28 views

Solution of Sylvester Equation with known principle diagonal

What is an easy method of finding solution to Sylvester equation of the form: $XA-AX = 0 $, for known $A$ and known principle diagonal of $X$?
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2answers
54 views

Find basis for orthogonal complements

Given V = $\mathbb P_{2}$($\mathbb {R}$) with an inner product space defined by $\mathbb <p,q> = \int_{-1}^1 \! p(x)q(x) \, \mathrm{d}x.$ Find a basis for $U^\perp$, where U = {rx | ...
6
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3answers
308 views

A gap in Halmos' definition of dimension? And how to repair?

In Halmos' Finite-Dimensional Vector Spaces, section I.8 has a proof of the Steinitz exchange lemma, which says that if $V$ is a vector space, $S$ is a finite independent subset of $V$, and $T$ is a ...
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3answers
56 views

Calculate Matrix A from eigenvalues, but no given eigenvectors

Here is my question: Write down a nontriangular 3 by 3 matrix whose eigenvalues are 6, 9, 2. I understand that you can calulate Matrix A using the formula A=V$\Lambda$$V^-1$, but is there a way to ...
2
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0answers
54 views

$T$ is an operator on inner product space, how to prove $T$ is invertible iff $T^*$ is invertible?

If $T$ is an operator on inner product space, how do we prove that $T$ is invertible iff $T^*$ is invertible? Can I change the goal to prove $T$ is injective iff $T^*$ is injective?
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1answer
80 views

Orthogonal Decomposition => Orthogonal Complements

Hi there can someone prove or disprove the hypothesis: $$\left(X=U\underline{\oplus}V\right)\Rightarrow\left(U^\bot=V,V^\bot=U\right)$$ (I don't require the space to be complete though)