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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Grade 8 simple algebra equation help

I find this question hard, please help It is given $x+(1/x)=3$ and $x^2+(1/x^2)=7$ Please find the value of $x^3+(1/x^3)$ Please show the steps
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Uncountable “relatively independent” subset of finite dimensional vector spaces over an uncountable field

Let $V$ be a $n$ dimensional vector space over an uncountable field ; then does there always exist an uncountable subset $S$ of $V$ such that any $n$ vectors of $S$ are linearly independent ? ( I can ...
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27 views

Weird contradiction between equations

A guy that I tutor came to me with the following question: The time it takes for body $A$ to pass 160 km is 5 hours longer than the time it takes for body B to pass 90 km. The speed of body A is ...
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1answer
13 views

Inversion of a matrix in a system of linear inequalities

I would like to know if someone knows sufficient conditions on $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^{n}$ such that for all $x\in\mathbb{R}^{n}$: $$Ax\leq b \Rightarrow x\leq A^{-1}b \text{ ...
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2answers
33 views

Proof with subspaces [on hold]

Prove: If $V$ and $W$ are three-dimensional subspaces of $\Bbb R^5$, then $V$ and $W$ must have a non-zero vector in common. (Hint: start with bases for the two sub-spaces, making six vectors in all) ...
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29 views

minimal polynomial of linear transformation

Let V and W are finite dimensional vector space over R.$ T_1:V\to V$ and $ T_2:W\to W$ be linear transformation whose minimal polynomials are given by $ f_1(x)=x^3+x^2+x+1 , f_2(x)=x^4-x^2-2$. Let $ ...
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1answer
36 views

$M$ is a faithful $A$-module.

Let $A = M_2(\mathbb{C})$ and let $M = \mathbb{C}^2$ with its usual $A$-module structure. How do I show that $M$ is a faithful $A$-module? I understand that this amounts to showing that its ...
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3answers
45 views

Does there exist a module structure over $\mathbb{C}^2$ as a $\mathbb{C}[x]$-module?

Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?
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2answers
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Module $M$ is infinite dimensional as a $\mathbb{C}$-vector space.

Let $M$ be a $\mathbb{C}[x]$-module. Since $\mathbb{C} \subset \mathbb{C}[x]$, we may consider $M$ as a $\mathbb{C}$-vector space. If $M$ is faithful, must $M$ be infinite dimensional as a ...
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Exists an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?

Let $M$ be an $R$-module with some torsion element $m$. Does there exist an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?
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32 views

What is the name for $Q A Q^T$?

Given square matrix $A$, very often we encounter the form $Q A Q^T$ for some square matrix $Q$. Is there a name for this form? I had been calling it a quadratic form, but it seems that term only ...
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22 views

Isolated Eigenvalue

What does it mean that an eigenvalue is "isolated"? My intuitive understanding says it is when one can find an open ball around it such that there is no other eigenvalue in that open ball. However, I ...
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3answers
39 views

Matrix having to be orthogonal, knowing it's norm-preserving

If we know that a real $m \times m$ matrix $C$ is norm-preserving, $||C\textbf{v}|| = ||\textbf{v}||$, then $C$ has to be orthogonal. Why should this be the case?
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convert the inverse of sum of two hermitian matrices into sum of two or more matrices.

I want to convert the inverse of sum of two hermitian matrices into sum of two or more matrices. I mean I want to simplify the bellow equation in a way that not to have inverse of sum of matrices any ...
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2answers
17 views

tangent for 3-dimensional function?

How can I calculate a tangent at a point $(x_0, y_0)$ in the direction $(r_1, r_2)$ for a $3-$dimensional function $f(x,y)$? I thought: \begin{equation*} T: (x_0, y_0, f(x_0,y_0)) + k \cdot (r_1, ...
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1answer
15 views

Transformation of parameter between the hyperbolas $xy=1$ and $x^2-y^2=2$ during rotation?

It is fairly straightforward to see that the hyperbola $xy=1$ is simply the hyperbola $x^2-y^2=2$ rotated by $\pi/4$. All we do is apply the corresponding rotation matrix to the vector ...
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21 views

Check for basis of a matrix

Given the matrices in $M_{3,3}$. ...
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22 views

Method to solve equation to find the value of each variable [on hold]

I am not very knowledge but maths but i have tried Gaussian Elimination but it didn't work $$mhe+fhe+mhs+fhs+mde+fde+mds+fds = 36$$ $$mhe+mhs+mde+mds = 17$$ $$fhe+fhs+fde+fds = 19$$ $$fhe+fhs+mhe+mhs ...
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2answers
23 views

Do the spaces spanned by the columns of the given matrices coincide?

Reviewing linear algebra here. Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \qquad ...
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1answer
23 views

What is the dimension of $f[x]$ over $f$

Let $f[x]$ be the ring of polynomials in one variable $x$ over the field $f$ with the relation $x^n =0$, for some fixed $n \in \mathbb{N}$. How can I find the dimension?
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Suppose$V_1,\ldots,V_m$ are vector spaces. Prove these two vector spaces are isomorphic [on hold]

The two vector spaces are: $L(V_1×\cdots ×V_m,W)$ and $L(V_1,W)\times\cdots\times L(V_m,W)$. Where $L(V,W)$ denotes the set of all linear maps from $V$ to $W$. Please help me with a rigorous proof, ...
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2answers
27 views

Solution to $A = BX + YC$ where $A$ is a square matrix of rank $n$, $B,C$ known, rank $m<n$

I hope this isn't too trivial of a problem. I'm really struggling with it and I feel like it shouldn't be that difficult. As stated in the title: Given (full rank): $A\in\mathbb{R}_{nxn}$ ...
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double integration with the same variable

I have the integral that I want to resolve. To calculate the flux of the electric machine, I have the following formula: $v_s= R_s \cdot i_s + \frac{\Phi _s}{dt}$ where $v_s, i_s, \Phi _s$ are ...
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1answer
15 views

Relationship between centralization and floating-point arithmetic

Suppose I have a problem of least squares where I have $n$ independents regressor variables $x_i$, and suppose I applied the process of centralization and scaling to this variable through ...
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2answers
18 views

Determine if the given set is a real linear space

Determine if the given set is a real linear space under the usual operations. The set of polynomials of degree $\leq 2$ The set of continuous functions on $[0,1]$ that satisfy $f(0)=0$ The set of ...
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25 views

What are Products and Quotients of vector space used for?

I'm self-studying the book Linead Algebra Done Right, and I'm now confused by these two space. What's the motivation for defining them? What are they used for? What's the insight of them? Please help ...
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28 views

Is there a relation between Ill-posed problems and Eigenvectors.

One can easily explain an ill-posed problem with an equation AX=b. The following link is an good example: http://www.encyclopediaofmath.org/index.php/Ill-posed_problems 1) Can there be a class of ...
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what's the potential application of low rank approximation of stochastic matrices

Suppose we have a stochastic matrix $P$ for a Markov chain, and we can compute a low rank approximation of $P$, say $P_k$, or we can find the nonnegative matrix factorization of $P$, i.e., $P=AW$ ...
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28 views

Proving that a Linear Transformation of a Subspace is a Subspace

I am having some trouble proving this: My attempt for the first part is to construct a corresponding system Ax = b Then check to see if this system is closed under addition and scalar ...
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2answers
35 views

Evaluate determinant of an $n \times n$ matrix, help

I need help with this problem: $D_{n}= \begin{vmatrix} 1 & 1 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 1 & 1 & \cdots & 0 & 0 & 0 \\ 0 ...
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2answers
15 views

Norm Used in Perturbation Matrix Thoery?

My question is that what is the type of 2-Norm used in Weyl's theorem for relative perturbation? Is that a induced norm, or a entry-wise norm? $\epsilon=\|X^T X-I\|_2$, where relative difference in ...
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1answer
13 views

Determinant and eigenvalues of Gram matrix lower bounds

I'm trying to find a non-zero lower bound on the determinant of the Gram matrix $\Gamma$ assigned to linearly independent set of vectors (is there such a lower bound?). But that is not my question ...
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1answer
22 views

How to find all lines with equal distance to two points in 2D space?

In 2D space: I've been given two points a, b and need to find all lines, that have same distance to these two points. From my ...
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1answer
13 views

vector space constructed through a torsion module

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form: $$ M = R/(p^{e_1}) \oplus \dots \oplus R/(p^{e_t}). $$ Let now be $_pM = ...
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1answer
19 views

a bilinear form is always the sum of two others

Let $K$ be a field with a characteristic, other than 2. Let $V$ be a finite dimensional vector space over $K$, and let $\gamma: V \times V \to K$ be a bilinear form. I now want to show that there ...
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3answers
45 views

Perpendicular vectors in 3d

Suppose a vector $v$ in $\mathbb{R}^3 $ How can I find two arbitrary unit vectors $u$ and $u^*$, that are perpendicular to each other and $v$ ? There are infinitely many solutions, but I cannot ...
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Finding the Eigenvalue of a general transformation

I'm studying for my Linear Algebra final and I'm having some issues with this proof: Given $T:V\rightarrow V$ $T=T^2$ what are all the possible Eigenvalues for T I'd appreciate any help you care ...
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1answer
34 views

Cholesky algorithm

Good afternoon everyone, I'm in need of a factoring algorithm cholesky and algorithms to solve upper and lower triangular systems, but I'm not finding any work in that octave. Recalling that need the ...
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1answer
11 views

Tensor and Hom are bi-adjoint functors for finite dimensional vector spaces?

Let $U$, $V$ and $W$ be finite dimensional vector spaces over a field $\mathbb{F}$. It is well known that $U\otimes_\mathbb{F}-$ and $\mathrm{Hom}_\mathbb{F}(U,-)$ are adjoint functors in the sense ...
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1answer
13 views

Orthogonal complex matrices: polar decomposition

Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)? thanks ...
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1answer
40 views

Rotation Matrix and programming [on hold]

I am actually programming in Android. An android tablet as a lot of sensors including one that gives the rotation vector of the tablet. (See ...
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0answers
27 views

Solving linear equation for low rank matrices

Consider $Ax=b$ where $A$ is invertible, so we have $x=A^{-1}b$. Now, let's consider a low-rank approximation of $A$, say $\bar{A}$ such that $rank(\bar{A})\leq r$ and $||A-\bar{A}||_F\leq \epsilon$ ...
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1answer
23 views

Matrix of Killing form a Lie algebra

Let $L$ be the Lie algebra with basis $B = \{u,v,w\}$, with $[u,v] = w, [v,w] = u, [w,u] = v$. Question : Find the matrix of the Killing form $\kappa$ of $L$ with respect to $B$. I have come across ...
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Relations between Kernel and image [on hold]

Let $T:V \to V$ be a linear transformation. Prove that $T^2=0$ if and only if $\operatorname{Im}(T) \subset \operatorname{Ker}(T)$.
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Linear transformations and conditions

Let $T: \mathbb{R}^n\to \mathbb{R}^n$ be a linear transformation given by \begin{equation*} T(x_1, x_2,...x_n)=(a_1x_1, a_2x_2,...,a_nx_n). \end{equation*} a) Under which conditions on ...
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1answer
37 views

Prove that the dual -space of the dual-space of V is isomorphic to V without using bases

Given a vector space $V$ the dual space $V^*$ is the space of all linear operators from $V$ to $\mathbb{C}$. $V^*$ is itself a vector space and I know how to prove $V \cong (V^*)^*$ by using a ...
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3answers
38 views

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$?

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$? My thinking is: $\langle u+v+w,v \rangle =0 \iff \langle u,v \rangle + \langle w,v \rangle = -1$ How do i ...
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2answers
27 views

eigenvalues of A - aI in terms of eigenvalues of A

I am stuck with this question of my assignment where given that A is nxn square matrix and a be a scalar it is asked to - Find the eigenvalues of A - aI in terms of eigenvalues of A. A and A - aI ...
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20 views

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$?

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$? Obviously $$N(z_1+z_2)=\left(\sum_{n=1}^{\infty} ...
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3answers
44 views

Least Squares method and Octave/Matlab [on hold]

I'll try to be as clear as possible so that you understand what I'm trying to do and can help me I have twelve pairs of data $(x_1,y_1),....,(x_{12},y_{12})$ and from this data we established a model ...