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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Why $(f\mapsto f(v_i)w)_{i,j}$ with $f\in V'$,$w\in W$ is a basis of $\mathscr{L}(V',W)$?

I'm trying understand the proof of the Proposition 3.1.2 (pg.5) of this document: http://www.win.tue.nl/~amc/ow/lba/lba3.pdf Suppose $V$ and $W$ are finite dimensional. If $(v_i)_i$ is a basis of ...
4
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1answer
120 views

Uniqueness of determinant

In Artin Algebra 2nd edition page 22, the author proved the uniqueness of determinant by saying that any matrix $A$ can be written in reduced row-echelon form $A'$: $A'=E_1\cdots E_kA$ where $E_i$ are ...
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10 views

Differentiable vector space valued functions doesn't depend on basis?

Differentiable vector space valued functions. Let $V$ be a vector space over $\mathbb F^n$ ($\mathbb R$ or $\mathbb C$) and let $v_1, \ldots, v_n$ be a basis for $V$. Define the linear isomorphism ...
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1answer
24 views

Expressing a $SL_2(\mathbb{R})$ matrix as product of…

If $\begin{bmatrix} a&b \\ c&d \end{bmatrix}$ is some matrix in $SL_2(\mathbb{R})$, then how can we express it as a product of matrices of the following type: $$\begin{bmatrix} s&0 \\ ...
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1answer
76 views

Eigenvalues of a Product of two matrices A and B inside trace operator expressed in terms of any eigenvalue of A or B?

This question has been in asked in a few varieties here but not in this one. If we have a real, symmetric, positive-definite matrix $A$ and a real, symmetric, positive-definite matrix $B$ and we know ...
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0answers
59 views

Why the space of all permutations of a vector (n!) is smaller than the space of all possible permutations of a sorting network?

Imagine you have a vector with 2048 entries. The total permutations are 2048! Now you have a sorting network let us say AKS, the total number of possible results with nlog(n) gates is $2^ {n log (n)}$ ...
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1answer
30 views

Clarification of some doubts: working with the restriction of a quadratic form

Let $q:\mathbb{R^3}\to\mathbb{R}$ such that $$q(x,y,z)=2x^2+3y^2+4xy-2xz.$$ I have to determine rank and signature of $q$, and so far it should be fine: I got $\operatorname{rk}(q)=3$ and ...
1
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1answer
23 views

Matrix associated with a bilinear form

We have $$b(v,w)=\begin{pmatrix} x_v& y_v& z_v \end{pmatrix} A \begin{pmatrix} x_w \\ y_w \\ z_w\\\end{pmatrix},$$ (where $A$ is the matrix associated with the bilinear form $b$ defined on ...
2
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1answer
19 views

Logic supporting column operations on matrices

In matrices, we justify row operations by drawing parallels with solving a system of equations i.e.: 1.Interchanging rows = Interchanging equations \ 2.Adding one multiple of a row to another = ...
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1answer
22 views

Visualizing cross product of points in 3-Space

If $p_0, p_1, p_2$ are three distinct points in space, then what does the cross product $$n = (p_0 - p_1) \times (p_0 - p_2)$$ mean geometrically? I'm having a little trouble visualizing this in ...
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1answer
13 views

Determining matrix M from $Mx_1 = b_1$ & $Mx_2 = b_2$, where $x_1, x_2, b_1, b_2$ vectors?

I have 4 vectors in the plane, $x_1$ and $x_2$, $b_1$ and $b_2$, and I'm told that there is a matrix $M$ such that $Mx_1 = b_1$ and $Mx_2 = b_2$. If I have a vector $x_3$, how do I determine $Mx_3$? ...
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2answers
42 views

Orthogonal complement $V^\bot$ of the vector space $V=\langle(1,0,2),(3,-1,0)\rangle$ and $V\cap V^\bot$

Consider the inner product defined by polarizing the quadratic form $$q(x,y,z)=x^2-z^2+4xy-2yz$$ on $\mathbb{R}^3$. Let $V=\langle(1,0,2),(3,-1,0)\rangle$. Could you show me how to find $V^\bot$ and ...
2
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2answers
95 views

Reference Request on a good Linear Algebra book [duplicate]

So I'm looking for a linear algebra book with a strong focus on proofs. It would be great if the book also uses concepts from regular abstract algebra like isomorphisms etc instead of dancing around ...
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0answers
15 views

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
17 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. ...
2
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1answer
35 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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0answers
41 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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3answers
258 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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53 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
42 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
24 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 $C$, ...
2
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0answers
34 views

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 ...
0
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1answer
92 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
29 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
45 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 ...
2
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1answer
37 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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1answer
62 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 ...
1
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1answer
27 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
34 views

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

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 $A=0$?
2
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1answer
27 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
43 views

Gradient function for restricted likelihood with respect to terms that influence Sigma

Is there a straightforward/generalized way to calculate partial derivatives for the gradient of the restricted multivariate log-likelihood function $\ln\mathscr{L}=C+\ln\lvert ...
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0answers
32 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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1answer
72 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 ...
0
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1answer
45 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
44 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
65 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
39 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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0answers
52 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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0answers
70 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
88 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 ...
6
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1answer
154 views

If $A^3B = BA^3$, then $AB = BA$.

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
51 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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1answer
60 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 + ...
1
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1answer
43 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 ...
4
votes
1answer
122 views

Linear Transformation on $\mathbb{R}^6$

Let $W$ be a vector space over $\mathbb R$ and let $T:\mathbb R^6 \to W$ be a linear transformation such that $S = \{Te_2, Te_4, Te_6\}$ spans $W$. Wich one of the following must be true? ...
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3answers
42 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 ...
0
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2answers
72 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 ...
0
votes
2answers
21 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? ...
1
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2answers
38 views

Product of matrices of a linear operator and of its inverse

Why if a linear operator $\phi$ is an isomorphism and we multiply the matrices of $\phi$ and of $\phi^{-1}$ in any basis, the result is $E$ the matrix that has $1$ on the diagonal and everything else ...
3
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2answers
80 views

Skew symmetric $4\times 4$ matrix of full-rank

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