Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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finding the derivative of g' and h'

So I know i'm not too terribly far off of the wrong answer but i'm not sure where I went wrong so I was just looking for a little help here. and sorry ahead of time but I don't know how to use the ...
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1answer
15 views

What is the change of basis in 2D?

I know how to apply a change of basis in 1D, but I was wondering: If I want to apply a change of basis to a nxn matrix, is it enough to apply the change of basis to every column of the matrix or is ...
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0answers
14 views

Bounding the inverse of a diagonally dominant matrix entry-wise

I have a $d \times d$ matrix $A$ whose entries are bounded (C1): $I - \epsilon X \preceq A \preceq I + \epsilon X$, where $I$ is the identity matrix and $X = 11^\top - I$ is the matrix with ones ...
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72 views

Row reduction and the characteristic polynomial of a matrix

Can you row reduce the matrix before computing $\det(\lambda I-A)$? Will this still give an equivalent characteristic polynomial?
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27 views

Calculate angle betwen two lines

I have been trying to find the best solution to this problem, but my math is pretty bad. What I want to do is calculate the "Angle" in radians, I have all the 3 co-ordinates and all the 3 lengths ...
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26 views

Basis of Trigonmetric Polynomials Help

Write the following trigonometric polynomials in terms of the basis functions: $\cos^2(x)$ $\cos^2(x) \sin^3(x)$ Is there a certain way to solve these types of problems because I'm very unsure on ...
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1answer
24 views

Do the given vectors span $\mathbb{R}^3$?

Do the following vectors span $\mathbb{R}^3$: $$v_1 = (2, -1,3)$$ $$v_2 = (4, 1, 2)$$ $$v_3 = (8, -1, 8)$$ I use Gaussian Elimination to bring the matrix to an echelon form, with a pivot of "1" in ...
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37 views

Find the coordinates of a vector (7,14,-1,2) in the basis (1,2,-1,-2), (2,3,0,-1), (1,2,1,4) and (1,3,-1,0) [on hold]

Not really sure what the basis (1,2,-1,-2), (2,3,0,-1), (1,2,1,4) and (1,3,-1,0)
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32 views

$(2,1+\sqrt{-5}), (1-\sqrt{-5},2)$ generate the $\mathbb Z[\sqrt{-5}]$-module $\langle 2,1+\sqrt{-5} \rangle \times \langle 2,1+\sqrt{-5} \rangle$

Ok, boring question here (I guess, at least). Let $R=\mathbb Z[\sqrt{-5}]$. Let $M=\langle 2,1+\sqrt{-5} \rangle$ the $R$-module generated by $2$ and $1+\sqrt{-5}$. I am asked to show that $M \times M ...
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Change of basis from Chebyshev to monomial basis for polynomials

I'm not that familiar with Chebyshev polynomials, so I hope I'm not too far off. Suppose that I have three order pairs $(x_0, f(x_0))$, $(x_1, f(x_1))$, and $(x_2, f(x_2))$ where $f : \mathbb{R} \to ...
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2answers
43 views

Form a basis for R^3? [on hold]

This is a homework problem and I need help on. Consider the matrix with the given vectors as its columns. Do (1, -1, 3), (-1, 5, 1), (1, -3, 1) form a basis for R^3?
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2answers
16 views

Proving a set is a basis for a subspace

The set $\{u_{1},u_{2}\cdots,u_{6}\}$ is a basis for a subspace $\mathcal{M}$ of $\mathbb{F}^{m}$ if and only if $\{u_{1}+u_{2},u_{2}+u_{3}\cdots,u_{6}+u_{1}\}$ is also a basis for $\mathcal{M}$. So ...
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2answers
19 views

Linear Algebra proof with column space

If $A$ and $B$ are two $m\times n$ matrices, then the column space of $A$ is contained in the column space of $B$ if and only if $A=BC$ for some $n\times n$ matrix $C$. So far I have that the rank of ...
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1answer
29 views

Show that Sine is not in the span of Cosine

Show that $\sin(x)$ is not in the span of $1$, $\cos(x)$, $\cos(2x)$, $\cos(3x)$, and $\cos(4x)$. Can I do this without Taylor series?
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24 views

Show that $F = \{a + b\sqrt{5} | a, b ∈ \mathbb Q\}$ is a field

Question: Show that $F = \{a + b\sqrt{5} | a, b ∈ \mathbb Q\}$ is a field under the operations - addition and multiplication where addition is given by: $(a + b\sqrt{5})+(c + d\sqrt{5}) = (a + ...
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0answers
10 views

why is the covariance matrix of a bekk model always positive definite?

The BEKK(1,1) model is given by: $$\Sigma_{t}=A_{0}A_{0}'+A_{1}a_{t-1}a_{t-1}'A_{1}'+B_{1}\Sigma_{t-1}B_{1}'$$ where $a_{t}$ are serially uncorrelated, zero mean innovations, $A_{0}$ is a lower ...
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1answer
17 views

Coordinates of a vector under a basis in a Hilbert space?

Given an arbitrary basis $\{m_1, \dots, m_n \}$of a Hilbert space $H$ (or just think it as $\mathbb R^n$, and I think the methods should be the same) with given inner product, how can we find the ...
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2answers
31 views

Show that a unique matrix exists for the coordinate vectors in a vector space

If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space $V$, there exists a unique matrix $M$ such that for any $f\in V$, $[f]_A=M[f]_B$. My textbook uses this theorem ...
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0answers
19 views

Find the eigenvector for an operator on a linear span

Let $V$ be the linear span of the functions $1,cos(x),sin(x)$. Let the operator $T$ on $V$ be given by the rule $Ty(x)=y(x+ \pi/4)$. Find the eigenvalues and eigenvectors of T in V. I know how to ...
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1answer
41 views

Proofs for $n$-dimensional vector spaces $V$

Suppose $V$ is an $n$-dimensional vector space. Prove that there is at most $n$ linearly independent elements in $V$. Prove that a set of $m<n$ element in $V$ cannot span $V$. I'm not really ...
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1answer
33 views

Matrix raised to a power

Find $A^n$ for $n = 1,2,...$. Does $A^n$ tend to a limit? $$A= \begin{pmatrix} 4/5 & 2/5 \\ 1/5 & 3/5 \end{pmatrix}$$ I found the eigenvalues $\lambda=1,2/5$ and the eigenvectors ...
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1answer
16 views

One Note about One to one and Surjective of linear functional [on hold]

I read a note that: if $ f \neq 0$ is a linear functional on H, then f is onto (surjective) and it is not one to one (injective) in general. Why this is true? i think it need advance ...
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2answers
13 views

Find basis of 4x3 Matrix

I've been confused with the following matrix. I'm trying to find the basis of the image of A: $$A= \left( \begin{array}{ccc} 6&4&10\\ 4&-1&3\\ -2&-5&-7\\ -10&-3&-13 ...
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27 views

Proving that a set of functions is a vector space

We've given that $V$ is a vector space and that $L(V)$ the set with functions $T:V\rightarrow \mathbb{R}$ s.t. $T(a_1f_1+a_2f_2)=a_1T(f_1)+a_2T(f_2)$. We must show that $L(V)$ is a vector space. I ...
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0answers
35 views

Isomorphism of vector spaces

Let $S$ be the space of all $3\times k$ matrices,$T$ be the space of all column vectors consists of seven components.If $S$ is isomorphic to a subspace of $T$ then what are possible values of $k$? I ...
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13 views

Difference between homogenous and nonhomogenous linear systems?

What is the difference between these two? My book doesn't give any explanation at all? I don't understand why a homogenous equation Ax=0 has a nontrivial solution iff the equation has one free ...
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How can y be used to factor N [duplicate]

Let x be the given muliple of φ(N). Then for any g in Z*[sub N] we have g^x=1 in Z[sub N]. Alice chooses a random g in Z∗[sub N] and computes the sequence ...
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1answer
25 views

How many ordered bases can be found for $\mathbb{Z}_p^n$ over filed $\mathbb{Z}_p$?

Take $\mathbb{Z}_p^n$ as a linear space over $\mathbb{Z}_p$. Now you can imagine multy bases for this space. (please leave a comment or have an edit if question is not clear enough.)
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1answer
22 views

Find a basis for the subspace of polynomials of degree 3

Let $\mathbb{P}_{3}$ be the collection of all polynomials of degree at most 3. Find a basis for the subspace consisting of those polynomials $p$ such that $p(1)=0$.
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2answers
57 views

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true?

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true? $1.$ $A=I$ $2.$ $\det(A)=1$ $3.$ $\operatorname{trace}(A)=n$ I have counter example for the first option.For ...
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1answer
16 views

Invertibe matrix is a transition matrix?

It is true that all transition matrices are invertible, but does the converse hold: All invertible matrices are transition matrices? I'm asking with regard to matrices over a field, but more general ...
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0answers
17 views

Division by 0 when solving linear equation using FFT with block circulant matrix

For a problem of Ax=b, if A is block circulant, let "a" be the first row of A, the problem is the same as circularconv(a, x)=b, therefore, x=ifft(fft(b)./fft(a)). However if I try a toy example with ...
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0answers
15 views

Use the Kronecker delta matrix to answer question

So I have the Kronecker delta which is denoted as $\delta_{ij}$=$I$. Let $b_1, b_2, \cdots, b_n$ be a set of $n$ real numbers, I must show that: $\sum\limits_{i=1}^n b_i \delta_{ij} = b_j$ and ...
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0answers
24 views

Is there any simple way of finding a matrix which commutes with a given (say, more complicated) matrix?

Suppose I want to find the eigenvectors and eigenvalues of a hermitian matrix $A$, but $A$ is big and ugly. Is there an easy way to find another, nicer, hermitian matrix $B$, such that $AB=BA$ and so ...
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1answer
80 views

True or False: Basis in the space of polynomials of degree less or equal to 2014 should contain polynomial of degree 2013. [on hold]

Basis in the space of polynomials of degree less or equal to 2014 should contain polynomial of degree 2013.
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1answer
19 views

Linear Spanning Functions [on hold]

Let $V$ be the linear span of the functions $1$, $\cos(x)$, $\cos(2x)$, $\cos(3x)$, and $\cos(4x)$. Is the function $\sin(x)$ in $V$? Justify your answer.
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1answer
42 views

Basis of the matrices with only non diagonalizable matrices

Is it possible to find a basis of $M_n(\mathbb{R})$ that only has non diagonalisable matrices ? I'm looking for a rather easy example, or a proof of the (non-)existence.
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25 views

What does it mean when a matrix row reduces to the identity matrix?

If a matrix row reduces to the identity matrix, what does that mean? The kernel is 0 vector? or basis for kernel is {}? Anything else?
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1answer
22 views

Proving that Mx is an eigenvector of B with λ as the eigenvalue given that $B=MAM^{-1}$ and $Ax=λx$

The square matrix A has λ as an eigenvalue with corresponding eigenvector x. The non-singular matrix M is of the same order as A. Show that Mx is an eigenvector of the matrix B, where $B = ...
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42 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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1answer
26 views

Why does Givens rotation avoid iteration and Jacobi rotation doesn't in case of reducing a symmetric matrix to tridiagonal?

I am currently implementing symmetric matrix reduction to tridiagonal. I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is ...
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30 views

statements of matrix analysis

Let $y$ be fixed value. Let $A=a(x,y)$ be a matrix and $f_{t}(x)=\frac{\sum_{n=0}^{\infty}{a^{(n)}(x,y)(\frac{1}{t})^n}}{\sum_{n=0}^{\infty}a^{(n)}(y,y)(\frac{1}{t})^n}$ Show that ...
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2answers
34 views

Can Cayley-Menger Determinant Be Negative?

Cayley-Menger determinant is used to calculate the area of a triangle, volume of a tetrahedron etc. Can be seen here. My question is; If given only positive numbers, can Cayley-Menger determinant ...
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0answers
19 views

If $\limsup_{t\to \infty} \int_{0}^{t}Tr(A(s))ds = \infty$ then $\limsup_{t\to \infty} |x(t)|=\infty$

For a homogeneous linear system of differential equations: $x'=Ax$ : Suppose that $\limsup_{t\to \infty} \int_{0}^{t}tr(A(s))ds = \infty$ ($tr(A):=$ trace of the matrix A). Then there exists solution ...
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1answer
21 views

subsapce of f(T)V T -invariant

On a vector space $X$, choose a nonzero element $v \in X$ and a linear map $T : V \to V$. $f(T)v$ is the space generated by $v, T(v), T^2(v),\dots$ I think any subspace of $f(T)v$ is also ...
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0answers
30 views

Determine matrix from linear transformation

Let $T_{1}$ and $T_{2}$ be linear transformations given by $$T_{1}([x_{1}, x_{2}])=[3x_{1}+5x_{2}, 4x_{1}+7x_{2}]$$ $$T_{2}([x_{1}, x_{2}])=[2x_{1}+9x_{2}, x_{1}+5x_{2}]$$ Find a matrix A such that ...
4
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1answer
39 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
1
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1answer
22 views

How to determine the Jordan form and give a Jordan base for a matrix?

given is $\begin{pmatrix} 3&0&-1&0&0 \\ 1&3&0&1&0 \\ 0&0&3&0&0 \\ 0&0&0&3&0 \\ 0&0&0&0&-3 \end{pmatrix}$ I have to ...
0
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1answer
21 views

Skew symmetric Matrix - Commutative property

If A and B are two odd size skew symmetric matrices(for example $3 \times 3 $). Let us say $X=AB,Y=BA$ Question What is the general relationship between X and Y? Can we write Y using X?
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1answer
48 views

Different Definitions of Tensor product, Halmos, Formal Sums, Universal Property

In the classic Finite-Dimensional Vector Spaces by P. Halmos he defines the Tensor product as The tensor product $U \otimes V$ of two finite-dimensional vector spaces $U$ and $V$ (over the same ...