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 R(A) = ker(A^t)?, where R(A) is the space generated by the columns of A

I'm looking at this deduction of the normal equations that solve the linear least squares problem. It goes like this: R(A) is the space generated by the columns of A $\hat{X}$ is the solution of the ...
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1answer
9 views

Prove: If $\Gamma$ is a collection of subspaces that is totally ordered by set inclusions, then the union of all members of $\Gamma$ is a subspace.

I have been mulling this problem over in my mind for the last couple days and I am stuck. There must be some basic principal I am missing. Closure with respect to scalar multiplication is obvious. ...
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0answers
10 views

Does negative definiteness imply anything about ALL principal minors?

Unfortunately I haven't received any response for my previous question, so I'm trying to solve it in a different way. I know that iff matrix $H$ is negative definite, its leading principal minors ...
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1answer
19 views

the values of K in the following system

I was assigned some homework for selfstudy but i cant make head nor tails of it. the assignment: for wich values of k in the following system does the system have no solutions one solution ...
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0answers
10 views

How to understand the meaning of 'Oblivious' in Oblivious Subspace Embedding?

For the definitions of Oblivious Subspace Embedding and Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf.
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11 views

Projection operator is Hermitian

Use Dirac notation (the properties of kets, bras and inner products) directly to establish that the projection operator $\mathbb{\hat P}_+$ is Hermitian. Use the fact that $\mathbb{\hat ...
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0answers
11 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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1answer
19 views

Recover the inverse after interative solution of a linear system

I have solved the linear system $\mathbf{A} \mathbf{x} = \mathbf{b}$ with an iterative solver. The problem is well-posed ($\mathbf{A}$ is invertible, $\mathbf{b} \ne \mathbf{0}$, blah blah blah). ...
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1answer
11 views

Vector notation for sum over elementwise product of 3 vectors

If I have an expression for two vectors $A$ and $B$ as below: $$\displaystyle \sum_{i=1}^N A_i B_i $$ we can write this as $ A^T B $ or $B^T A$ Now, if I have 3 vectors $A$, $B$ and $C$, ...
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18 views

Norm of a linear function [on hold]

$f : V \to V$ is a linear function. The basis of $V$ is $v_1, v_2, v_3$ Suppose $$\begin{align} f(v_1) &= 3v_2 \\ f(v_2) &= -5v_2 \\ f(v_3) &= 2v_1\end{align}$$ The norm of all basis ...
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Rank of two bases

Suppose V and V' are finite spaces and A is the matrix of $\phi$ of whatever of two basis of V and V'.prove r($\phi$)=r(A).Now if we have basis $e_1 ... e_n $ the rank(A) is equal to the columns or ...
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1answer
31 views

Proving Submultiplicativity on a Matrix Norm

Let $||A||=(\sum_{i=1}^{n}\sum_{j=1}^{n}{a_{ij}^p})^{1/p}$, and let p=2. Then prove that $\|AB\|\le \|A\|\|B\|$ I have looked at numerous proofs for this, and I don't see one that satisfies me ...
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2answers
15 views

Requirement That a Vector be Related to Itself Through Identity

If I have two vectors for which the relation can be written $$ \begin{bmatrix}\vec{I}_1\\\vec{I}_2\\\vec{I}_3\end{bmatrix} = \begin{bmatrix}A\end{bmatrix} ...
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2answers
16 views

Invariant subspace - simplified definition

I just, just, started reading about invariant subspaces, but I don't think I'm getting a really concrete idea of what they are. Could someone try to explain to me more advanced examples of this? This ...
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1answer
24 views

Derivations on matrix algebra

Let $M=M_2(\mathbb{C})$ and let $\delta:M \mapsto M$ be a $\mathbb{C}-$linear map such that $\forall a,b \in M$ we have $\delta(ab)=\delta(a)b+a\delta(b)$. Prove that $\delta$ is of the form ...
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0answers
17 views

R in QR decomposition always upper triangular? [on hold]

Why is the matrix R in a QR decomposition always an upper triangular matrix?
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1answer
25 views

Linear algebra. Can I find this combination?

$ \alpha_f f + \alpha_h h = a \\ \alpha_g g + \alpha_h h = b \\ $ I think I should be able to find $f+h$ and $g+h$ as functions of $\alpha_i, a, b$ but I can't manage.
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1answer
31 views

Extending Minkowsky inequality to double summation?

I know the Minkowski inequality for sequences as follows : $$\left(\sum_{k=1}^n|x_k+y_k|^p\right)^{1/p} \leq \left(\sum_{k=1}^n|x_k|^p\right)^{1/p}+\left(\sum_{k=1}^n|y_k|^p\right)^{1/p}$$ Now say we ...
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1answer
17 views

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ with characteristic polynomial $-\lambda(\lambda-3)^2$ and $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ be a diagonalizable endomorphism with characteristic polynomial $-\lambda(\lambda-3)^2$ such that $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$. Given these data, ...
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1answer
21 views

Symmetric real matrix $A$ with $0$ as the only eigenvalue, does this imply $A=0$?

Let $A$ be a real symmetric $n \times n$ matrix ($n$ a positive integer). Let $0$ be the only eigenvalue of $A$. Does this imply $A=0$?
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18 views

Basis existance of kernel [on hold]

If we have basis of $V$ that consists of $n$ vectors and if $e_1,e_2,\ldots,e_k$ is basis of $\ker\phi$, why numbers $x_1,x_2,\ldots,x_k$ exist such $x_{k+1}e_{k+1}\cdots x_ne_n=x_1e_1\cdots x_ke_k$ ...
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0answers
15 views

Distance between all rows in 2 matrices expressed as a matrix equation

I have two matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. All real numbers. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element ...
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1answer
13 views

Least squares approximation problem of $t^3$ in a subspace spanned by even degree polynomials.

I am having trouble solving the following question, Let $P_9 ([-1,1])$ be the complex vector space consisting of polynomials $p:[-1,1] \rightarrow\mathbb{C}$ with degree 9 or lower. With the inner ...
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0answers
15 views

Eigenvalues of (restrictions of) the standard representation of $S_n$

Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible. Is there any way to understand ...
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0answers
33 views

How can I find the common axis of 2 cones in space that have the same base radius but different heights?

How do I find the 3D vector describing the axis of 2 overlapping cones, like this: If I have only the following information: Coordinates of the common tip Coordinates of a point on the yellow ...
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1answer
21 views

Quadratic to matrix form

How can I show that $J(w) = \Sigma_{i=0}^m u_i(w^Tx_i-y_i)^2$ can be re-written to $J(w)=(Xw-y)^TU(Xw-y)$ and how can I differentiate the 2nd equation with respect to w? where $x_i \in\mathbb {R^n}, ...
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1answer
33 views

Eigenvalue of multiplication in a number field.

Let $\mathbb{K}$ be an algebraic number field. $b \in \mathbb{K}$ defines a linear transformation $\phi(b)$ on $\mathbb{K}/\mathbb{Q}$ by multiplication - $\phi(b)(x):=bx$ for all $x\in\mathbb{K}$. ...
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1answer
19 views

find the distance between line and point R3 [on hold]

I would like to know how I can find the distance between the line and point in R3 the equations of line: $$ \left\{ \begin{array}{c} 2x+y+z=2 \\ 3x+4y-z=3 \end{array} \right. $$ and point (3, ...
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1answer
37 views

What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
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1answer
26 views

Proving a transformation is a linear transformation

If I have a transformation $T:V \to V$ given by $T(f(x)) = x*f(x)$. To prove this would I just show that $T(a*f(x_1) + b*f(x_2)) = a*T(f(x_1)) + b*T(f(x_2))$ to show addition and scalar multiplication ...
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2answers
31 views

to find the determinant of a matrix given the roots of an equation [on hold]

Given an equation $x^2 +2x +2=0$ and $a$ and $b$ is the root of this equation. Find the determinant of This matrix \begin{bmatrix} a & 1 & 0\\ 0 & b & 1\\ 1 & b & 0 ...
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0answers
46 views

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices? [duplicate]

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices or having to choose a particular base ? Such a proof should probably use a non matricial definition of traces. One ...
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1answer
26 views

Expectation of matrix product

Suppose we have a random matrix $M \in \mathbb{R}^{n\times m}$ such that $\text{E}[M] = 0$ and $\text{E}[M M^\top] = \Sigma$. How does one compute $\text{E}[M^\top M]$?
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4answers
42 views

How are standard basis of polynomials linearly independent? [duplicate]

Consider the set $\{1,z,z^2,...z^m\}$. As this is the standard basis for a vector space of polynomials, the list should span the space and also be linearly independent. One problem I'm having though ...
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0answers
14 views

Prove that enumerable set of complex exponentials is linear independent

Define $f_j(p) = e^{i u_j \cdot p}$ for $j=1,2,3,...$, $u_j, p \in \mathbb{C}^N$, $i = \sqrt{-1}$ and $\cdot$ is the scalar product. I need help to prove that the set $\{f_j : j=1,2,...\}$ is linearly ...
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0answers
10 views

Has any system a minimal subsystem [on hold]

Has any system a minimal subsystem? I can't really grasp the concept, could someone maybe provide an example?
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1answer
88 views

Show that A and B commute

Let $A$ be a Hermitian matrix. Suppose there exists a matrix $B$ such that $A^3B = BA^3$. Show that $AB = BA$. I was trying to use the fact that since $A$ is Hermitian, there exists a unitary ...
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0answers
25 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. Any hint would be appreciated! Thanks ...
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0answers
22 views

Formula to calculate angle on a fan or semicircle

How do I calculate the angle shown in the picture given the height, width, and the arc deduction of $2$? I had applied the Right Triangles formula to calculate the hypotenuse: $h^2 = a^2 + ...
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1answer
34 views

Is it a subspace or not?

Is $$\mathscr{S_1}=\lbrace(a,b,c)\in\mathbb{R}^3:a^3=b^3\rbrace$$ subspace of R^3? my try:: $a^3=b^3\iff a=b$ in $\mathbb{R}$. So $\mathscr{S_1}=\lbrace(a,a,c)\in\mathbb{R}^3\rbrace$ this is a plane ...
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1answer
26 views

Linear Transformation on R^6 [on hold]

I am stuck on this linear transformation problem.Thanks for the help!
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2answers
32 views

Linear Algebra. Closed unit ball. Prove. [on hold]

Hi, dear all, how can I prove that? Please help me. I need your help very much!
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0answers
40 views

Give me an idea [on hold]

We've got an infinite number of cards, each of them having a positive integer written on it.Prove that however we choose 2015 cards, having the sum of the numbers written on them 4028, we can divide ...
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3answers
32 views

Proving a transformation is not a linear transformation

I'm asked to prove if a transformation is linear or not. In the vector field $V=\{f(x)\colon \mathbb{R} \to\mathbb{R}\}$, so the transformation is $T\colon V \to V$ given by $T(f(x)) = (xf(x))+1$. I ...
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2answers
42 views

Finding Eigenvalues of given linear operator

Find the eigenvalues and the eigenvectors of the linear operator $T:C^\infty(0, 1)\to C^\infty(0, 1)$ $T(f)(x) = \frac{f'(x)}{x}, x \in (0,1) $ Using the definition : $TF = \lambda F \iff ...
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2answers
17 views

Is the rank of a matrix unaffected by congruence transformations?

Sorry for the easy question but if I have a square matrix $A$ over $\mathbb C$ then is its rank invariant under a congruence transformation $A \mapsto P^t AP$ ? What's the easiest way to see this? ...
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2answers
21 views

Linear operator matrix

Why if a linear operator $\phi$ is an isomorphism and we multiply the matrices of $\phi$(x).$\phi^{-1}(x)$ in any basis, the result is E the matrix that has 1 on the diagonal and everything else is 0. ...
2
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2answers
27 views

Skew symmetric 4x4 matrix of full-rank

I have come across the fact that a 4x4 skew-symmmetric matrix of full-rank is equivalent to \begin{pmatrix} 0 &\theta_1& 0 &0 \\ -\theta_1& 0 &0 &0 \\ 0& 0&0 & ...
2
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2answers
22 views

Linear transformation with special properties

how should I do that please (I had this in my test yesterday)? Linear transformation $f:\mathbf{R}^{10} \to \mathbf{R}^7$ has an attribute that every vector $\mathbf{v}$ for which is true that ...
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1answer
20 views

invariant subspace under some conditions

This came out at my linear algebra exam and I was not able to solve it. Let $f\colon \mathbb{R^{3}} \rightarrow \mathbb{R^{3}}$ be a linear transformation such that $\langle f(u),f(v) \rangle = ...