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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What is the matrix $\left[ DS(A) \right]$, which gives $\left[ DS(A) \right] H=AH+HA$?

In Hubbard's multivariable calculus book $DS(A):H \mapsto AH+HA$ is introduced as a linear transformation where $A$ is an $n \times n$ matrix, $S(A)=A^2$, and $D$ is the notation for derivative. It ...
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16 views

Linear Operators and Convergence

I am struggling with this question from my Differential Equations course: If $T$ is a linear transformation on $\mathbb{R}^n$ with $||T-I||<1$, prove that $T$ is invertible and that the ...
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8 views

Nonsingularity of submatrices

I'm still working on my question: Warm start of simplex algorithm after update of constraint matrix. While reading Schrijver's book "Theory of linear and integer programming" (reprint 1999) I ...
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12 views

Check if the equality holds

I have the following problem. For orthogonal $8\times 8$ matrix $M$ ($M\cdot M^{T} = 1$) check if the following equality holds $$ U = M^{T} \cdot \left( \begin{array}{cc} 1_{3\times 3} & 0\\ ...
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1answer
12 views

Riesz representative of gradient of $f(u) = u^*u$ in different inner products

This is a seeming "paradox" that has been bothering me for some time, as it (or other situations like it) show up often when computing gradients for numerical optimization on complex vector spaces. ...
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Checking whether the result is positive definite or positive semi-definite with two methods

Given, $$A = \begin{bmatrix} 1 &1 & 1\\ 1&1 & 1\\ 1& 1& 1 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Using Method 1: ...
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12 views

Operator Norm of a Linear Transformation of a Matrix

The book I am using for the ODE course is Differential Equations and Dynamical Systems by Lawrence Perko. I am having a difficult time understanding what an operator norm of a linear transformation ...
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28 views

Prove that for a quadratic form $Q(x) = x^TAx$, $A$ is symmetric.

We want to show that $A=A^T$. Any matrix $A$ may be written as a sum of its symmetric and skew symmetric parts: $A = \frac{A+A^T}{2} + \frac{A-A^T}{2}$. Upon substitution, we obtain $Q(x) = x^T ...
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What is the good way to remember the signs of the rotational matrix?

Recall rotational matrix in (x,y) is given by: $R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$ For the life of me I cannot remember if the ...
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16 views

Transformations of Quadratic Forms to their Normal Forms

Assume we are given a quadratic form in one variable, $Q(x) = ax^2$, where $a \in \mathbb{C}$. We may construct a change of coordinates $x = cX$, where $c \in \mathbb{C}$ and is of our choosing. Upon ...
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Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
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2answers
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volume of a $n$-d parallelepiped with sides given by the row vectors of a matrix $A$ is the product of the singular values of this matrix $A$

Why the volume of a $n$-dimensional parallelepiped with sides given by the row vectors of a matrix $A$ can be seen as the product of the singular values of this matrix $A$? I only know in ...
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Matrices and Linear Algebra- Determine if the list is linearly independent in the real vector space. [on hold]

1.Determine if the list $((3,2,0,1),\,(2,1,4,0),\,(0,-1,12,-2))$ is linearly independent in the real vector space $\mathbb R^4$. 2.In the real vector space $C(\mathbb R,\mathbb R)$ of all continuous ...
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69 views

Showing an equation has one positive root

Let $n\geq 2$ be an integer and $\beta > 0$. Consider the polynomial equation: $$p(x) = x^n + x^{n-1} - \beta = 0$$ Show the equation had exactly one positive root $p(\beta)$ Do I use the ...
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25 views

Field of formal Laurent series over $F$.

Let $F$ be a field and let $K$ be the set of all functions $f\in F^\mathbb{Z}$ satisfying the condition that there exists an integer (perhaps negative) $n_f$ such that $f(i)=0$ for all $i<n_f$. ...
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35 views

Balancing chemical equations using linear algebraic methods

I know there are already plenty of questions on this site regarding this topic but I am having difficulty with a particular chemical equation. I am trying to balance the following: $$ { C }_{ 2 }{ H ...
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30 views

Then prove that $\exists U$ subspace of $V$ such that $W_i \oplus U=V$ $\forall i=1,…,n$

Let $V$ be finite dimensional vector space over an infinite field. Let $W_1, W_2,...,W_k$ are subspaces of $V$ of same dimension. Then prove that $\exists U$ subspace of $V$ such that $W_i \oplus U=V$ ...
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45 views

How many $2\times3$ real matrices are needed to guarantee that at least one of them is a linear combinations of the others?

The only thing I know is that $$\left(\begin{array}{ccc}1&0&1\\0&1&1\end{array}\right)$$ Seems to have a column to be linear combinations of the others.
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Linear Algebra help needed [on hold]

It's been a while since I've taken linear algebra and I am trying to figure out the problem below. I am not sure how to start. Thank you for any and all help. I am not sure how to For any real ...
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25 views

How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
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2answers
23 views

Proving the existence of an inverse of a matrix. (Linear algebra)

Suppose that $A$ has no inverse. Prove that there exists a vector $b$ such that $Ax = b$ has no solution My try Proving by contradiction , Assume that for all vector $b$, $Ax = b$ have at least one ...
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38 views

Proof for linearity on tensor products

Theorem: Let $U$ and $V$ be vector spaces. Let $\mathbf{u}^* \in U^*$. Define $\mathbf{f} : U \otimes V \to V$: $$\mathbf{f}\left(\sum_{r} \mathbf{u}_r \otimes \mathbf{v}_r\right) = \sum_{r} ...
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1answer
14 views

Choleski decomposition of a positive matrix

Let us consider a matrix $\boldsymbol{F}$. We consider its Choleski decomposition, $ \boldsymbol{F} = \boldsymbol{M} \boldsymbol{M}^T $. We know that $\boldsymbol{F}$ needs to be positive definite. ...
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Determinant structure of symplectic matrix

I want to show that if $\lambda$ is a real eigenvalue of a symplectic matrix $A$ then its char poly is of the form $\det(A-\mu id) = (\lambda-\mu)(\frac{1}{\lambda}-\mu) \det(\hat{A}- \mu id) $ where ...
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3answers
26 views

Determining the intersection of kernel and image.

I was posed the following question: If $T$ is a linear operator on a finite dimensional vector space $V$ such that rank of $T$ = rank of $T^2$. I'm supposed to show that the kernel and image of $T$ ...
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I need to write equations for 2 rates of change. [on hold]

here is my problem, Write the following as an equation. x / y 1 / 3 2 / 12 3 / 27 4 / 36 5 / 51
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19 views

Finding a variable in the determinant of sum of matrices [on hold]

I Don't Know how to earn P that is scalar from Below Formula : R = log2(abs(det(I + P * H*H'))) Everything is known except P. P is scalar and positive. I is an Identity NxN , H is complex NxN ...
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1answer
61 views

$(1)$ $a\boxplus b=a+b-ab$ for all $a,b \in F$. $(2)$ $a\boxdot b=1-t^{\log_t{(1-a)}\log_t{(1-b)}}$ for all $a,b\in F$

Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows: $a \boxplus b=a+b-ab$ for all $a,b \in F$. $a \boxdot b=1-t^{\log_t ...
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29 views

Measuring the effect of a linear transformation on the result

I have an unknown vector $x\in\mathbb{R}^n$, a known orthogonal matrix $\Phi\in\mathbb{R}^{n\times n}$, a known matrix $A\in \mathbb{R}^{m\times n} (m \le n)$, and a known vector $b\in \mathbb{R}^m$ ...
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1answer
11 views

eigenvalues of cycle graph and its complement graph

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...
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Need help about field decision; mathematics or physics? Who can be good at these? [on hold]

First of all, you may want to delete this question because it is not an mathematical question, but this question can be an opening door to thousands of mathematical question. Hello everybody, I need ...
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1answer
17 views

Finding jordan normal form

Let be $T:\mathbb{R}^7\rightarrow \mathbb{R}^7$ Such that $(T-15I)^3=0$ and $\dim\text{Im}(T-15I)^2=2$ find the Jordan normal form of $T$ If $(T-15I)^3=0$ so the minimal polynomial can be ...
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1answer
37 views

Is $\left \{ x_{1}+y_{1},…, x_{n}+y_{n}\right \}$ a basis for $\mathbb{R}^{n}$?

Suppose $\left \{ x_{1},..., x_{n}\right \}$ and $\left \{ y_{1},..., y_{n}\right \}$ are two different bases for $\mathbb{R}^{n}$. Is $\left \{ x_{1}+y_{1},..., x_{n}+y_{n}\right \}$ also a basis for ...
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3answers
24 views

Linear Independence and Subset Relations

I've been reading the wikibook on Linear Algebra and in the section 'Linear Independence and Subset Relations' it defines the following lemma: Lemma 1.14: Any subset of a linearly independent ...
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1answer
24 views

An orthogonal matrix has eigenvalue $1$ with the eigenspace $E(1)$ of dimension $n-1$. Then $-1$ is also an eigenvalue with $E(-1)$ of dimension $1$?

Let $(V,g)$ be an $n$-dimensional Euclidean space ($g$ scalar product) and let $f:V \to V$ such that $g(f(u), f(v)) = g(u,v)$. It is known that the matrix associated to $f$ with respect to an ...
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1answer
19 views

Show that a set of polynomials are linearly independent in the complex space

I have been trying the solve the following question without any success: Let $\lambda_1, \lambda_2, \lambda_3$ be three distinct complex numbers and define the polynomials $m(\lambda), m_1(\lambda), ...
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1answer
13 views

Question about forced damped oscillators

This question is from my linear algebra book: Find the function $f(t)$ of the form $f(t)=a \cdot cos(2t)+b \cdot sin(2t)$ such that $f''(t)+2f'(t)+3f(t)=17cos(2t)$ All I've figured out so far is ...
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1answer
23 views

Notation and name for this function?

Let $k \geq 1$; let $V,W$ be vector spaces; and let $T: V \to W$ be linear. Then how do we call and denote the function $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?
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Looking for the basis of the kernel of T

Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : $P_2$ → $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
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How do i show V is a linear subspace if it's defined like this?

Let $V =\{(x,y,z) \in \mathbb R^3 : x+3y=3z\}$,and let $T :V \to \mathbb R^3$. be given by $T(x,y,z)=(x,y,z)\times(1,3,−3)$, the usual cross-product in $\mathbb R^3$. How do i show that V is a ...
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1answer
28 views

For any linear operator $\phi$ on $V$, prove such an integer $m$ exists.

Suppose $V$ is an $n$-dimensional vector space over some infinite number field $K$, $\phi\in\mathcal L(V)$, prove there exists such a (positive) integer $m$ that $$\text{Im} \phi^m=\text{Im} ...
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51 views

Show that $Ax=0, Bx=0$ share the same solution space iff there is some invertible $P$ s.t. $B=PA$.

The question is said in the title, suppose $A,B\in M_{m\times n}(K)$, where $K$ is some infinite number field. If we regard $A,B$ as linear maps from $K^n$ to $K^m$, then they share the same ...
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Solving a system of ODEs with 4 repeated eigenvalues

I'm working on problem which requires me to solve a system of ODEs with 7 equations. I've gotten as far as determining the eigenvalues and vectors of my coefficient matrix $A$, but 4 of the ...
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How do you show that f(z)=z conjugate isn't linear?

let $x_1= a+ib,x_2= c+id,k=$scalar $f(x_1,x_2)=f(x_1) + f(x_2)$ $f(a+ib + c + id)=(a+c)-i(b+d)$ $f(a+ib)+f(c+id)=(a+c) - i(b+d)$ $f(kx_1)=kf(x_1)$ $f(k(a+ib))= k(a-ib)$ $kf(x_1)=k(a-ib)$ Looks ...
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1answer
9 views

Minimum of restricted linear combinations.

Let $\{N_0, ... , N_m\}$ be a set of natural numbers, then the minimum $(\geq 1)$ of all their linear combinations is their GCD. Is there a way to calculate that minimum if some $N$s can only be ...
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Given the tetrahedron $OABC$, find a condition on $a OA+ b OB + c OC$ such that this is always inside $ABC$.

I did the following: Taking the tetrahedron $OABC$, one can decompose it in: $OA,OB,OC, AB,BC$. And then, writing: $$x(BC-AB)+AB\quad x\in[0,1]$$ We obtain all the points in the line segment from ...
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1answer
41 views

I want to appeal this problem from an exam in Linear Algebra I, do you think its appealable? [on hold]

I have the follow question : Let $U_1, U_2, W$ are linear spans of linear space $V$ while V is finite. Proof: If $$U_2 \cap W \neq \{0\}$$ $$U_1\cap W\neq \{0\}$$ $$U_1 \cap U_2=\{0\}$$ Then $dimW ...
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1answer
27 views

Complex conjugate of $z$ as a different variable

Can a complex conjugate be represented by a different letter than $z$? As in: Let $y$ be a complex number satisfying $|y|<1$. Find the set of all complex numbers $z$ satisfying ...
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1answer
40 views

What is a change of basis and how do i find it?

W is a four dimensional vector space over a field F with basis S = (v1, v2, v3, v4). B is also a basis of W such that. $b1 =−v1, b2 =v1 +v2, \, b3 =−v1 −v2 −v3, \, and \, b4 =v1 +v2 +v3 −v_4.$ ...