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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22 views

3 equations 4 unknowns.

I need to know what does this system describes please: $x+3t=2$ $y+t=-1$ $z+2t=1 $ Now subtracting equation (1) from (2) and (3) i.e (1)-(2)-(3) leads to $x-y-z=2$ hence the set of points of the ...
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0answers
6 views

If w is a 2-vector in char not 2, such that $w \wedge w \not = 0$, why is w decomposable?

I want to see this as directly as possible, since I want to understand the variety of lines in P^3. I don't want to use the general convolution formula (given in Shaferevichs Varieties book). Any ...
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1answer
32 views

What is the meaning of “Continuous Group ”?

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" I did not understand the meaning of this statement " The method is based on finding a nontrivial continuous group ...
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1answer
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 ...
3
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0answers
30 views

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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1answer
25 views

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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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. ...
3
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2answers
94 views

How to solve this system of nonlinear equations?

How to solve these equations for $a$, $b$, $c$ and $x$? I have the following: \begin{align} 1 &= 2a+b+c\\ a &= (a+b)x + 0.25(a+c)\\ a&=(a+c)(1-x)\\ b&=a(1-x)+c(x-0.25)\\ ...
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3answers
955 views

How prove this matrix have a positive eigenvalue

let $$A=\begin{bmatrix} a_{1}&a_{2}&a_{3}\\ a_{4}&a_{5}&a_{6}\\ a_{7}&a_{8}&a_{9} \end{bmatrix}$$ where $a_{i}>0$, show that the matrix $A$ At least one positive ...
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0answers
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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2answers
43 views

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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29 views

The highest direction of the trace operator

Let $W$ be a real and symmetric matrix ${m \times m}$ from the set $M_{m,m}$, and $T:M_{m,m} \rightarrow \mathbb{R}$ a function defined by $T(W) = trace(W^3)$. We are interested to find the ...
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1answer
21 views

Isolate Costs in NPV equation

Hey can anyone help with this? This is the classic NPV equation: NPV = -CapEx + ∑ (Revenue − Costs) / (1+Discount)^i The partial sum is from i = 0 to n years. For my purposes all the elements ...
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1answer
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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0answers
60 views
+50

compare norms on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
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2answers
19 views

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: ...
2
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4answers
31 views

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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1answer
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 ...
3
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0answers
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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0answers
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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2answers
15 views

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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1answer
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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0answers
19 views

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 ...
3
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3answers
93 views

Multiples of determinant are elements in a matrix

Suppose we have an $n \times n$ square matrix, $A$. Let the determinant be $|A|.$ We also constrain the elements of $A$ such that each element of $A$ is an integer multiple of $|A|.$ Is there an ...
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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 ...
9
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1answer
1k views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
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1answer
390 views

general formula for an orthogonal projection of a point onto a line

Could someone confirm this or correct the mistakes because this seems somehow wrong although I double checked it. $(m_x,m_y)$ are coordinates of a point , $(p_x,p_y),(k_x,k_y)$ are coordinates of a ...
3
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2answers
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 ...
3
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1answer
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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0answers
35 views

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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0answers
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 $i<n_f$. ...
3
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1answer
31 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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2answers
38 views

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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0answers
17 views

Optimizing a multivariate quadratic fn

Let $D\in \mathbb{R}^{m\times n}$, where $m\geq n$, and D is full column rank. I'd like to find $\sup_{x\in \mathbb{R}^m}f(x)$, where $f(x):=\frac{-1}{2}x^T DD^T x+c^T x$. I know the answer is: ...
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0answers
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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0answers
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 ...
3
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2answers
51 views

Bound on maximum angle between vectors

I have two vectors $\mathbf v_1$ and $\mathbf v_2$: $$\mathbf v_1 = \begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}, \mathbf v_2 = \begin{pmatrix}x_2\\y_2\\z_2\end{pmatrix}$$ The components of these ...
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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. $b1 =−v1, b2 =v1 +v2, \, b3 =−v1 −v2 −v3, \, and \, b4 =v1 +v2 +v3 −v_4.$ ...
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0answers
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
29 views

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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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 ...
0
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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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1answer
39 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
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 $b\in \mathbb{R}^m$ ...
1
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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. ...
0
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3answers
27 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 $T$ ...
0
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1answer
75 views

Are rotations linear mappings?

Given a rotation matrix $\rho:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ with $det(\rho)=1$ and $c \in \mathbb{R}$ is it correct to say $\rho(cx)=c\rho(x)$?
4
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1answer
762 views

Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known

Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$. Let $D$ be a diagonal matrix with the same dimensions and rank. What is the fastest way to compute $(A+D)^{-1}$? ...
-1
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0answers
21 views

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