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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Proving determinants using properties of determinants [on hold]

$$\begin{vmatrix} 1 & a^2+bc & a^3\\ 1 & b^2+ca & b^3\\ 1 & c^2+ab & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a^2+b^2+c^2)$$ we have to solve this by using the properties of ...
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Can the coefficients used to prove a set of functions is linearly dependent be imaginary?

Example: $\cos x$, $e^{ix}$, $3\sin x$. I can show: $C_1\cos x + C_2 e^{ix} + C_33\sin x = 0$ if $(C_1,C_2,C_3) = (1,-1,i/3)$ But i don't know if $C_3 = i/3$ is a valid coefficient to choose. Can ...
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How to determine whether it is vector space? [on hold]

Does the set of all polynomials of degree exactly $5$, together with all the constant polynomials,determine a vector space?
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My first proof related to subspaces (vector spaces). Please comment.

What do you think about my first proof which deals with subspaces? Theorem An intersection of subspaces is a subspace. Preliminaries Corresponding to the notation in Wikipedia, symbols for vectors ...
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How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
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Linear Transformation $T-T^2=I$

Let T be a linear transformation from a vector space V over reals into V such that $T-T^2=I$. Show that T is invertible Solution: I started by multiplying T on both sides and getting $-T^3=I$
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Question about proof of Cauchy-Schwarz inequality.

I was trying to prove the Cauchy-Schwarz inequality, and came up with the following: $$ |u||v|\cos{\theta} \leq \frac{1}{2}|u|^2 + \frac{1}{2}|v|^2 $$ I got stuck here, did some googling and found a ...
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47 views

Quadratic Prime

We had received some questions on Quadratic equations, But I wasnt able to do one. Here it goes: Let $a,b$ be natural numbers $a>1$. Also, $p$ is a prime number. If $ax^2+bx+c=p$ for 2 distinct ...
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Generalized Schur complement theorem

Let $M$ be an $(n+m)\times(n+m)$ real non-symmetric positive semidefinite (PSD) matrix partitioned as \begin{eqnarray*} M=\left(% \begin{array}{cc} A~~B\\ C~~D\\ \end{array}% \right), ...
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Curves through points [on hold]

An astronomer wants to determine the orbit of an asteroid about the sun. He sets up a Cartesian coordinate system in the plane of the orbit with the sun at the origin. By Kepler's law the orbit must ...
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The nullity of a square matrix with linearly dependent rows is at least one. TRUE OR FALSE

Here is the answer my textbook gives. http://imgur.com/ycCRoWK I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
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32 views

TRUE OR FALSE: Matrices with linearly independent row and column vectors are square.

Here is the answer of my textbook: http://imgur.com/vEoY31O Why must a matrice with linearly independent vectors have nullity(A)=0? That is where I lose track of the question. Are zero rows ...
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3answers
72 views

$A^2=cA$ for some $c \neq 0$

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $$A^2 = cA.$$ Questions. 1) Is there a matrix class for matrices ...
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Modifying U=mxn SVD Algorithm to U=mxm Algorithm

I have painstakingly ported this Python source "svd.py" to C++. I confirm it works for the example it comes with. While testing another example (this one, from Wikipedia), the assert statement trips ...
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Vector Algebra-Direction Cosines [on hold]

If $\cos\alpha$, $\cos\beta$ and $\cos\gamma$ are the direction cosines of a straight line then find the values of $\cos2\alpha$, $\cos2\beta$ and $\cos2\gamma$.
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43 views

Can I say that a space of R³ is a subspace of R³?

I am dealing with some questionsm asking if some Set of variables are Vector Spaces of R³. My question is a simple one, a matter of interpretation (I couldn't find a clear answer). Asking if a Set is ...
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25 views

Prove that a normal matrix is unitary/Hermitian

I'm stuck with these two questions for while. I'd appreciate your help. ...
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Linear Algebra problem (related to transpose matrices)

Producing $x_1$ trucks and $x_2$ planes requires $x_1+50x_2$ tons of steel, $40x_1+1000x_2$ pounds of rubber, and $2x_1+50x_2$ months of labor. If the unit costs $y_1, y_2, y_3$ are \$700 per ton, \$3 ...
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1answer
24 views

complexity of solving $n \times n$ rank deficient linear system

I think it is known that given a nonsingular $A \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^n$, solving a linear system $Ax =b$ for $x$ can be done in $O(n^3)$ steps. Now assume $A$ is of rank ...
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2answers
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Number Of Solutions Of Homogeneous And Non-Homogeneous System

let there be a matrix $A^{n*m}$ that $Ax=b$ the solution set of the homogeneous system $H=(h\in F^m; Ah=0)$ the solution set of the non-homogeneous system $L=(l \in F^m; Al=b)$ How do |L| and ...
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31 views

Normal Matrix Having all real eigen values is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
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No Solution Set for augmented coefficient matrix $[A\mid b]$.

Let there be $Ax=b$ a set of m equations with n unknowns. If $\operatorname{rank}(A) \neq \operatorname{rank}[A|b] $ the system of equations does not have a solution Does it mean that: there is a ...
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Subspace in 2D Help

I understand that any line through the origin is a subspace of a vector space. Why won't a curved line through the origin be a subspace in 2D? When I think of a line through the origin, I think of a ...
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1answer
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Multiple Cartan sub-algebras

How is it that for a Semi-simple Lie Algebra there is not one Cartan Sub-Algebra? I assume since there are multiple CSA's of a SS Lie algebra that must mean some of the ss elements of the Lie ...
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Basis and Linear Transformation Proving Problems [on hold]

Can someone help me with these problems? I'm really poor at proving.
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Diagonalizabilty of ad(adjoint map)?

let $\mathsf{g}$ be a finite dimensional lie algebra and $\xi\in\mathsf{g}$. Under which conditions the adjoint map $ad_\xi :\mathsf{g}\longrightarrow \mathsf{g}$ is diagonalizable? what about ...
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Sum of Determinants = Scalar product with normal vector?

Today I have seemingly simple question and maybe someone knows the answer without getting into messy calculations. So we have $n$ vectors $v_1,\dots,v_n\in\mathbb{R}^n$ and let us assume for the ...
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Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
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An interesting linear algebra question

Let $A$ and $u$ be $n\times n$ matrix and $n\times 1$ vector of $\mathbb{C}$. Denote $\overline{A}$ is the matrix $(\overline{A})_{ij}=A_{ij}^*$, the conjugate number; ($\overline{A}$ is not the ...
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what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
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dimension of vector space such that $MN=0$

Consider the matrix $\displaystyle M=\begin{bmatrix} 1 &0 &-1 \\ 0 &1 &0 \\ 1 &1 &-1 \end{bmatrix}$ and let $S_M$ be the set of $3\times3$ matrices N such that $MN=0$ . ...
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What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
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Prove that function is inner product

$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$. I need to show this function is an inner product: $$\langle p,q\rangle=\sum_{j=0}^n ...
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How to find the point of intersection with three equations?

Given the following equations with three variables $a, b, c$ $a-5b+4c=-3$ $2a-7b+3c=-2$ $-2a+b+7c=-1$ How can I determine the point (if it exists) at which all three lines intersect?
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An elliptic element of $SL(2,\mathbb{R})$ conjugate to a rotation

Show that an elliptic element of $SL(2,\mathbb{R})$ is conjugate to a rotation. An element $A$ of $SL(2,\mathbb{R})$ is called an elliptic element if $|\text{tr}(A)|<2$ As $|\text{tr}(A)|<2$ ...
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Need verification - Given a Hermitian matrix and two eigenvectors corresponding to distinct eigenvalues, show x and y are orthogonal.

Claim: Let $A \in \mathbb{C}^{mxm}$ be hermitian ($A = A^*)$. If $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. Proof: Let $x$ and $y$ correspond to ...
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Kronecker product and the vec operator: confusion on proof of vec(AXB) = (transpose(B) ⊗ A) vec(X)

I was reading up on Kronecker products and vec operator from couple of sources and landed on the equation: vec(AXB) = (transpose(B) ⊗ A) vec(X) suppose ...
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1answer
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If $T$ is not diagonalizable over $\Bbb R$ and $T$ has all its eigenvalues real, then can I say $T$ is not normal?

$\Bbb V$ is a vector space of dimension $n$ and $T$ is a linear operator on $\Bbb V$ I know that if $T$ is not diagonalizable over $\Bbb C$ then $T$ is not normal. My question is if $T$ is not ...
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1answer
29 views

Don't understand Levi decomposition theorem

Levi decomposition theorem states that any finite-dimensional real Lie algebra $L$ is the semidirect product of a solvable ideal and a semisimple subalgebra. I don't understand this since to me it ...
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2answers
171 views

Show A and B have a common eigenvalue

Let A, B and C complex square matrices such that: $ C\neq 0 $ and $AC=CB $ prove that A and B has a common eigenvalue. It's worth mentioning that earlier in the assignment I have proved that ...
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Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
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The unit vector in the direction of u

I'm trying to work my way through a linear algebra assignment, and I'm struggling with a few questions. This is one of them. I'm completely lost at question A. I think that B's answer is true, but ...
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Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
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Finding the order of elements in a Galois Field

Does there exist a Galois field GF(4)? GF(4)={0,1,2,3}; If we take this Galois field, then the element '2' is not having any degree..? So is it possible to construct GF(4) ?
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Continuity in finding eigenvectors

I'm wondering whether there's "continuity" in the eigen vectors of different matrices corresponding to appropriate eigenvalues. For instance, if we change certain elements in a matrix, can we ...
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Choose $h$ and $k$ such that the system has, no solution, a unique solution, and many solutions.

Looking through my textbook, I see no examples as to how to solve this \begin{align} x - 3y & = 1 \\ 2x + hy & = k \end{align}
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Finding minimum point of a function using linear algebra

Given a function $$q(x,y)=2x^2-2xy +2y^2$$. Find the minimum point of the following function by first converting it to a matrix form and using the diagonalisation of the matrix to find its minimum ...
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Question about inner products

Given a real or complex vector space $\;V\;$ and a (finite) basis $\;B\;$ of it, does it always exist an inner product on $\;V\;$ s.t. $\;B\;$ is an orthonormal basis with respect to it? The question ...
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Group inverse of positive semi definite matrix

Group inverse and Moore Penrose inverse of a positive semidefinite matrix are same. How?
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For What $a$ The Linear Equations Have Sloutions

$2x+ay-z=-2$ $x-3z=-3$ $x+2y+az=-1$ I have thought about reducing a matrix so in the end I will have an equation with $a$ then I can determine for which $a$ the are one solution/infinite ...