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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relation between vector space and torsion free module

Please help me to know this.Let $R$ be a domain and let $A$ be an $R$ -module and $Q=Frac(R)$. then a module $A$ is a vector space over $Q$ if and only if it is torsion -free and divisible. thanks for ...
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3answers
45 views

Where does $\Lambda=P^{-1}AP$ come from?

How do we derive the fact that if a matrix is diagonalizable then we can diagonalize it with the formula $\Lambda = P^{-1}AP$, where $P$ is a block matrix whose columns are the eigenvectors of $A$? I ...
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1answer
17 views

Rank of a linear map $f: \mathbb{R}^n \to \mathbb{R}^m$ is equal to $m$?

Take a linear map $f: \mathbb{R}^n \to \mathbb{R}^m$, where $n>m$. Is the rank of $f$ always equal to $m$? Since the image of $f$ contains $\{f(a) | a \in \mathbb{R}^n \}$, the image will contain ...
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0answers
7 views

Gauss-Newton Non-Linear Squares Optimisation

I doubt this is solvable at all, but I thought I will give a try. Essentially I am trying to extend Gauss-Newton algorithm to 2nd Taylor term. ...
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1answer
11 views

Linear dependence under transformation

I have a linear map $f:\mathbb{R}^6\rightarrow\mathbb{R}^4$ and I'm asked to show that if $u$,$v$ and $w$ are linearly dependent vectors in $\mathbb{R}^6$ then also $f(u),f(v),f(w)$ in $\mathbb{R}^4$ ...
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1answer
13 views

The annihilator induces a module

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
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11 views

Homomorphisms inbetween factor modules

Consider the Ring $\mathbb{Z}$ and the two ideals $(n), (m)$, where $n, m \in \mathbb{N}$, and consider $GCF(m, n)$ (the greatest common factor of $m, n$). Let p: $\mathbb{Z}/(m) \to \mathbb{Z}/GCF(m, ...
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1answer
16 views

Find vector and parametric vector of a line

I have a line that is perpendicular to a plane. This perpendicular line is $3i-2j+6k$. I've also been given that the line passes through $A(2,3,0)$. I'm unsure on how to represent this line as a ...
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3answers
52 views

How do I find the determinants of $3A, -A, A^2, A^{-1}$, where A is an $4\times 4$ matrix and $\det(A) = \frac{1}{3}$?

I am getting crazy with these determinants. For a little, I thought I could solve a problem alone, because I had understood more or less how to calculate the determinants of a matrix, but I am back to ...
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1answer
16 views

Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
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1answer
44 views

What is the difference between $A^{-1}$ and $A^\Theta$?

Let $A$ be a square invertible matrix. Then $$A \cdot A^{-1} = I$$. Let $A^\Theta$ be the conjugate transpose matrix of $A$. Then $$A \cdot A^\Theta = I$$. Both on multiplication with $A$ gives ...
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0answers
24 views

Proving that a linear functional is matrix trace [duplicate]

Let $W=\operatorname{M}_{n\times n}(\mathbb{F})$ (square matrices $n\times n$ over $\mathbb{F}$), and $f\in W^*$. If $f(AB)=f(BA)$ for every $A,B\in W$ and $f(I)=n$ prove that ...
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8 views

How do I find a matrix for this linear transformation

The problem is that the transformation is defined by T(p)=(p(0), p(1) P(-1), p(0)) B is the standard basis for M22 and B' = {1, x, x^2}. How ...
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1answer
11 views

How to find linear equation from text

A company produces covers for cell phones. The costs of a production of $x$ units can be described by a linear function $C(x)=ax+b$. At a production of $150$ units the costs are DKK $6000$. At a ...
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3answers
19 views

Find the vector passing through a given point which is orthogonal to a given triangle in space

I'm given this problem where I have 3 points in space $A(3, -1, 2)$, $B(-2,1,2)$ and $C(2, 0, 5)$. I need to find the vector passing through point $A$ that is perpendicular to the triangle made by ...
2
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0answers
17 views

Change of basis and similarity

Consider the transformation $T$ in the standard basis: $$[T]_B\begin{bmatrix} 0&3&1 \\ -1&3&1 \\ 0&1&1 \end{bmatrix}$$ Also consider the two matrices: $$A_1 = ...
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3answers
29 views

$\dim (W_{1} \cap W_{2}) = \dim W_{1}$ implies $W_{1} \subset W_{2}$?

Let $V$ be a finite-dimensional vector space and let $W_{1}, W_{2}$ be subspaces of $V$. If $\dim (W_{1} \cap W_{2}) = \dim W_{1}$, must $W_{1} \subset W_{2}$? Since $\dim (W_{1} \cap W_{2}) = \dim ...
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0answers
21 views

Finding an Extermal of Hard Examples?

Who Can show me the calculation for solving extermal for $$\int_0^1 (x^2+ \dot {x}^2+2xe^t) dt \quad \text{ when }\quad x(0)=0,\;x(1)=free.$$ My TA say a short answer and I Couldn't reach to ...
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0answers
18 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
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0answers
5 views

generalized inverse and its applications

As generalized inverse has vast applications in the field of linear algebra, but why the generalized inverses is important? why we are studying about it?
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20 views

tensor of two vector space

I don't know how to show this problem please help me. let $R$ be a domain and $Q=Frac(R)$ if either $C$ or $A$ is a vector space over $Q$,prove that both $C\otimes_RA$ and $Hom_R(C,A)$ are also vector ...
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1answer
19 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
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1answer
11 views

Problem with Broyden update: Divide by a matrix?

I am implementing a maximum likelihood method (the EM algorithm) for which I'm using Broyden's method at each iteration. Here is the formula: $\Delta A = \frac{(\Delta \theta - A ...
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2answers
48 views

An example of non euclidean inner product [on hold]

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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0answers
19 views

The dual of the dual, isomorphism, equivalence of functions

Problem: Let $V$ be a f.d. vector space. Define $\theta:V\to (V^*)^*$ given by $\theta(v)(\alpha)=\alpha(v).$ Let $T:V\to V$ and $T^*\ ^*:(V^*)^*\to(V^*)^*$ be linear maps. Prove $T=T^*\ ^*$. ...
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1answer
14 views

Basis, polynomial vectors

Given the vector space $P_3(R)$, find a basis for it containing the polynomials $x^2 + 1$ and $x^2 - 1$. To find a basis, I need to find whether there exists constants in front of these two vectors ...
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2answers
30 views

Intuitive explanation why if $P$ is a subspace of linear space $L$, then $L/P$ is not a subspace of $L$

Is there an intuitive explanation of why: if $P$ is a subsppace of linear space $L$, then $L/P$ is not a subspace of $L$. I know that it is true, but it is counter intuitive to me.
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29 views

Which space it belongs to Hilbert, Banach or something else?

The question is related to the following two questions. The link: Understanding Eigenvector defines the problem at hand. The question is the in which space (Hilbert or Banach or something else) we ...
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0answers
14 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
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0answers
15 views

Induced map between quotient vector spaces well defined and linear?

Suppose we have vector spaces $V,W$ and subspaces $$ V\supset V'\supset V''\\ W\supset W'\supset W'' $$ suppose also that we have a linear map $A:V\to W$. What does it take for this map to induce a ...
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2answers
24 views

Is $(A \oplus B)^{\perp} = A^{\perp}\cap B^{\perp}$?

Is this true? $$(A \oplus B)^{\perp} = A^{\perp}\cap B^{\perp}$$ I am trying to prove this, but could not find a way. Any suggestions would be much appreciated. Thanks.
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1answer
36 views

Eigenvectors of derivative

I'm trying to consider how linear algebra relates to calculus. It seems to me that the only eigenvectors of the derivative operator on $\Bbb R$ are the functions $ce^{kx}$ for constants $c$ and $k$. ...
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1answer
18 views

Why do the 1's in Gauss Jordan RREF need to be along main diagonal and not other diagonal?

I've practiced G-J elimination and understand most of the algorithm insofar as it represents the different manipulations one can apply to a system of equations. However, when we're talking about ...
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1answer
21 views

Linear Algebra-Vector Subspaces Question [on hold]

Let $U=\{f\in C^1([-1,1],\Bbb R);y'=y+1\}$. Is $U$ a subspace of $C^1([-1,1],\Bbb R)$?
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28 views

Rates and Linear Equations [on hold]

The following question I found in an old high school textbook I bought in a second hand bookshop. The question is exactly as it appears in the text with no additional information. The answer, with no ...
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1answer
27 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
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0answers
11 views

Minimal column sum (?) solution to system of linear equations over $\mathbb{Z}_2$

There are $n$ equations in $n$ unknowns where both the coefficients and unknowns come from the field $\mathbb{Z}_2$. I can represent these as the equation $Ax = b$ where $A$ is an $n\times n$ matrix ...
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2answers
80 views

What space to use?

My apology if this question is not mathematical. I have heard of many spaces, Hilbert space, Banach space etc. But could not connect a specific problem to a space. For example if I ask a mathematical ...
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2answers
30 views

An orthogonal projection matrix in $ \Bbb{R}^{3} $.

Consider the vector space $\mathbb{R^3}$ with usual inner product. Find the orthogonal projection matrix on the xy plane. I've found sometimes the orthogonal projection of a vector in a given ...
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2answers
34 views

Is the sum of two projections a projection?

Let $ S $ and $ T $ be two linear subspaces of $ \Bbb{R}^{2} $. Then is the sum of the projections $ P_{S} $ and $ P_{T} $ (i.e., $ P_{S} + P_{T} $) a projection? I don’t think it is since the ...
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Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but ...
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1answer
22 views

Equation for the curve in terms of x,y

we got the equation $$r(t) = (t-2)i + (t^2+4)j$$ I got $$x = 1-2t$$ $$y = 1+4t$$ Would that be correct?
2
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49 views

What is the point of basis vectors?

Why do we even bother with basis vectors? Why don't we just notate an element $x$ of an $n$-dimensional vector space $V$ as an ordered set $(x_1,x_2,...,x_n)$ and go from there?
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1answer
34 views

For what values of a constant does the system have: No solution; More than one solution; Unique solution [on hold]

Consider the linear system . For what values of a does the system have: a) No solution; b) More than one solution; c) Unique solution It should be answered by augmented matrix.
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3answers
28 views

Linear algebra, inner product and matrix

Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric ...
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2answers
35 views

For which value of k will the vector be a combination of two vectors?

For which value of $k$ will the vector $\begin{bmatrix}1\\-2\\k\end{bmatrix}$ in $\mathbb{R}^3$ is a linear combination of the vectors $w=\begin{bmatrix}2\\-1\\-5\end{bmatrix}$ and ...
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0answers
9 views

Representation of Affine Maps

I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$: $$ f[\lambda x+ (1-\lambda) y]=\lambda ...
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5answers
75 views

Product of any two arbitrary positive definite matrices is positive definite or NOT? [duplicate]

Suppose that , $A$ and $B$ are $n\times n$ positive definite matrices and > $I$ be $n\times n$ identity matrix. Then which of the followings are positive definite ? (i) $A+B$ (ii) $ABA$ ...
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0answers
5 views

Find all vertices of a parallelepiped given only 3 to start with (linear algebra)

My question is simple. I just want to find out the rest of the vertices given only three of them. I haven't really grasped the process of finding the vertices, so I need someone to help me understand ...
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0answers
10 views

Find an orthogonal basis of inner product

Let's define dot procduct $<A,B>=Trace(A B^T)$ over $M_{n \times n}(\mathbb{R})$ Find basis or system of equations describing an orthogonal $W^\perp$ subspace to subspace $W$ which consist of ...