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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Proving full column rank of a matrix

Let $x$ be a $K\times 1$ vector of random variables satisfying that $E[xx']$ is nonsingular. For some given integers $M\geq 1$ and $L\leq K$, let $z_1,\ldots,z_M$ be $L\times 1$ column vectors ...
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Issues with a particular bilinear form and determining rank, signature, etc. of its restriction

Let $b: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$ such that $b(X,Y)=trace(X^tAY)$, where $X^t$ is the transpose of $X$ and $A=\begin{pmatrix} 2 & 1\\1 & 0\end{pmatrix}$. In my ...
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Kernel and diagonalizability of endomorphism $f:\mathbb{R_2[x] \to R_2[x]}$ such that $f(p)=p(1)x^2-p(k),$ for $k \in \mathbb{R}$.

Problem: Let $f:\mathbb{R_2[x] \to R_2[x]}$ be the endomorphism on the space of polynomials of degree less or equal than two such that $$f(p)=p(1)x^2-p(k),$$ for $k \in \mathbb{R}$. I have to ...
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Dimension of sum of permutations of tensor products of vector spaces

Sorry for the mouthful of a title! Suppose I have two finite vector spaces $W,V$ with bases $\{w_1\dots w_p\}$ and $\{v_1\dots v_q\}$. Consider some subspace $S$ of $W\otimes V$ of dimension $m$ ...
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Linear Transformation from $\mathbb R^n$ to $\mathbb R^m$: image of $1$ dimensional subspace has dimension $1$ or $0$ [duplicate]

I am struggling to comprehend the question below. Especially the meaning of 'the image of $L$ under $F$'. Let $F : \mathbb R^n \to \mathbb R^m$ be a linear transformation. Prove that if $L$ is a ...
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Unit balls and the Schatten norms

I have a very naive question: Let $A$ and $B$ $n \times n$ (complex) matrices with operator norms $\|A\| \leq 1$ and $\|B\| \leq 1.$ Pick a $1 \leq p < \infty.$ Then with a constant $K_p$ ...
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45 views

Eigenvalues of symmetric matrices are real without (!) complex numbers

Is there any proof of the fact that the eigenvalues of symmetric matrices (i.e. $A\in\mathbb{R}^{n\times n}$ with $A^t=A$) are real without the use of the concept of complex numbers?
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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 ...
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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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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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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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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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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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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 ...
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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 ...
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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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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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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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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 ...
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Car-van ratio problem [on hold]

There are 2/5 as many vans as cars and 2/3 as many motorbikes in a parking lot. What is the ratio of vans to the total number of cars and motorbikes?
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which one is non singular matrix [on hold]

let A,B be n-square matrices such that $$BA+B^2 =I-BA^2$$ where I is the n-square identity matrix . which of the following is always true 1. A is non-singular 2.B is non singular 3. A+B is non ...
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29 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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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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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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1answer
13 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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the values of K in the following system [on hold]

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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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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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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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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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
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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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25 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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36 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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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
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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
26 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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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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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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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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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
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Symmetric real matrix $A$ with $0$ as the only eigenvalue, does this imply $A=0$? [on hold]

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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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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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
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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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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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42 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
22 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
35 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
20 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, ...