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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Point-Slope Equation of a line. Why is one answer incorrect and other is correct?

I am reviewing basic algebra. I am using quiz from the link this, and I solved the equation on paper and I get answer which is showing incorrect, I do not understand why is it wrong? It says my ...
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0answers
12 views

For what $p$ is the condition number of a given matrix $A$, using the $p$ norm on matrices, minimal?

For what p is the condition number of a given matrix $A$, using the p norm on matrices, minimal? The condition number on $A$ is given as: $$K(A,p) = \|A\|_p \times \|A^{-1}\|_p$$ I tried ...
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2answers
35 views

Why is this a useful way to prove the characterisation of bases?

So I have come across the following theorem(see below)in my linear algebra notes and am slightly confused. I feel I understand both the statement and the proof, my confusion arises from the fact that ...
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0answers
27 views

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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1answer
23 views

inner product of positive semi definite symmetric matrices [on hold]

I have a positive semi definite symmetric matrix $X$, $(n\times n)$. let $X=vv^T$ s.t $\|v\|=1$. I came to a point where I am stuck to show which is: $v^TYv=\langle X,Y\rangle$ (How to show this ...
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32 views

Changing the subject of a formula

How do i make the subject of the formula $$A=\frac{m}{n}+\frac{n}{k}$$ into k? Also, does anyone know a website that solves this type of question for you? Mathway works fine, though I don't have ...
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2answers
28 views

Linear combination of basis function in logarithm space

I have a function $f(x)$. As theory said that it can represent by linear combination of basis functions such as $$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$ where $\alpha$ is coefficient and $g(.)$ is basis ...
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1answer
23 views

What condition do I have set to have $x_j$ for $j=1,…,n$ to be non-negative?

I have $\sum_{j=1}^n q_{jj}(x_j-y_j)^2\le1$ What condition do I have set to have $x_j$ for $j=1,...,n$ to be non-negative? The book I am reading says $\sqrt{q_{jj}}\ge1/y_j$ but why?
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1answer
41 views

Jordan canonical form of an upper triangular matrix

Find the Jordan canonical form of the matrix. Justify your answer. $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{bmatrix} $ My Try: The eigenvalues ...
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0answers
26 views

Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
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1answer
28 views

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$.

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$. My approach: Suppose span$\{u_1,\dots,u_n\}=W$, where$\{u_1,\dots,u_n\}$ is a set ...
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37 views

$C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.

Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid ...
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0answers
9 views

No nontrivial invariant subspaces iff characteristic polynomial is irreducible

Say $V$ is a nonzero, finite dimensional vector space over $F$, and $T\in \mathcal{L}(V)$. I want to show that the only $T$-invariant subspaces of $V$ are trivial iff $f_T$, the characteristic ...
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0answers
20 views

Constructing a periodic function around $f(t): R \to R^3$

Definitions: $f(t): R \to R^3$ $\hat{\bigtriangledown}f(t) := \frac{df(t)}{dt} \frac{1}{|\frac{df(t)}{dt}|}$ $\vec{a}e^{\vec{b}x} := \vec{a}\cos(x) + \vec{b}\sin(x)$ $\vec{a}, \vec{b} \in H$ Is ...
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2answers
95 views

Looking for a gentle intro to Linear Algebra

Does anyone know of any gentle, introductory books to LA that assume little prerequisites, even in the way of vectors and matrices? I want something that will give intuition and reasonable proofs, and ...
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4answers
85 views

why does the reduced row echelon form have the same null space as the original matrix?

What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?
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2answers
45 views

Basis of a product vector space

Let $E$ a vector space of dimension $p$ with $(e_1, \ldots, e_p)$ as a basis. Define the cartesian product vector space $F = v_1^\top E \times v_2^\top E \times \ldots \times v_n^\top E$ where the ...
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13 views

Project a signal $S(t) = \sum_0^{\infty}A(k)e^{if(k)t}$ to 3d domain $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $

Definitions: $\vec{v}e^{\vec{w}x} = \vec{v}\cos(x) + \vec{w}\sin(x)$ $ \psi_0(t) = x\hat{i}t + y\hat{j}t + z\hat{k}t, \ x,y,z\in R$ $ \psi_{n+1}(t) = \psi_n(t) + \hat{v}_nA_ne^{\hat{w}_ntf_n} $ ...
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2answers
62 views

If some power $A^n$ of a matrix $A$ is symmetric, is $A$ necessarily symmetric?

If $A^{n}$ is a symmetric matrix, should I conclude that A is also symmetric?
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0answers
23 views

Can this be expressed in terms of linear constraints?

I'm attempting to find a matrix $X$ that minimizes some function $f(X)$ subject to the constraint that $$ X=W A Z $$ where $A$ is a given non-negative matrix with rows that sum to 1, and $W$ and $Z$ ...
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0answers
13 views

Conditional expectation of a set of Gaussian variables

I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ? Specifically: For a pair of Gaussian random variables $[x,y]$, the ...
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1answer
23 views

In what condition we have $(K^{-1})^\ast = (K^\ast)^{-1}$?

Suppose $X$ $Y$ are two finite dimensional Hilbert space. Assume $K$: $X\to Y$ is linear. My question is, in what condition of $K$ that $$(K^{-1})^\ast = (K^\ast)^{-1}?$$
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1answer
18 views

Geometric and Algebraic Multiplicity, zero dimensions

The eigenvalues are $\lambda =0$(because we have multiplication here), $\lambda =1$, and $\lambda =2$ for the given characteristic equation, and as (a) states, that $GM\le AM$. Now, I want to know ...
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1answer
23 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
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1answer
24 views

Different representations of a matrix in reduced row echelon form

EDIT: I decided to ask this question after working on this particular problem. I had the stupidity to think that row-reduced = row reduced echelon form. Brain fart, nothing more to see here... ...
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0answers
24 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
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16 views

Maximum in a linear system of equations

I have a system of equations with a tridiagonal coefficient matrix: $$ \alpha_i f_{i-1} + f_{i} + \beta_i f_{i+1} = \Gamma_i $$ , where $i$ goes from 1 to $n$. For a given $M$, what constraints ...
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2answers
30 views

About the elements of a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$

Let $G$ be a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$. I would like to prove that for every $g \in G$, $\det(g) \in \lbrace -1,1 \rbrace$. Here are my ideas : since $G$ is a finite subgroup ...
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1answer
18 views

Matrix norm and perturbation problem on finite dimensional $V$

Suppose we have a finite dimensional real vector space $V$ equipped with a norm $\|\cdot\|$ given by $$\|x\|^2=x^tXx$$ where $X$ is a matrix and $x\in V$ is in matrix form. Further let us assume that ...
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2answers
32 views

Final transformation matrix

I have a 3d object, to which I sequentially apply 3 4x4 transformation matrices, $A$, $B$, and $C$. To generalize, each transformation matrix is determined by the multiplication of a rotation matrix ...
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1answer
22 views

Matrix inequality $A^2 \succeq A$

If $A$ symmetric positive semidefinite matrix is the following inequality true. If $A \succeq I$ then \begin{align} A^2 & \succeq A \end{align} This is an equivalent of $a^2 \ge a$ is $a \ge ...
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31 views

Prove $A,B$ share an eigenvalue [duplicate]

Let $A, B, C \in M_n(\mathbb{C})$ (not zero matrices) and let $g(x)\in\mathbb{C}[X]$. Let's assume $AC=CB$. Prove that $A,B$ share an eigenvalue. Things I've already proved (followed by the ...
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2answers
37 views

Finding matrix representation of an Ellipsoid [on hold]

I have a $2$-dimensional ellipsoid centered at $(1,2)$. The axes are parallel to $y=x$ and $y=-x$, and it passes through points $(-1,0)$, $(3,4)$,$(0,3)$,$(2,1)$. I would like to find the symmetric ...
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1answer
21 views

Linear, Squared and Logarithmic scales with given input domain and output range

The input domain is $[12,24]$ and the output range is $[0,720]$. Now I know that with using linear scaling the value $16$ of the input range is mapped to $240$; with using sqrt scaling the same value ...
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0answers
26 views

Rotating one coordinate system about another

I have two coordinate systems: A and B. I also have a point p, whose position relative to ...
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4answers
75 views

Prove that $g(A)$ is an invertible matrix

Let $A\in M_n(\mathbb{C})$ and let $\lambda\in\mathbb{C}$. Prove that if $\lambda$ is not an eigenvalue of $A$ then $A-\lambda I$ is invertible. Moreover, for $g(x)\in \mathbb{C}[x]$, prove that if ...
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1answer
29 views

AX=B in Matlab solution [on hold]

I am new to MATLAB and wanted to solve a linear equation and come across a problem The matrix A is [ 3 6; 6 14] and B Is [ 5;11] According to ...
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3answers
53 views

How can I find the co-ordinate of where a line intersects a circle?

I was looking to know if there was an equation that would allow me to calculate the co-ordinates of a point on the circumference of a circle where a line intersects it and the center. My diagram ...
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4answers
1k views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
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1answer
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Show that every vector in the null space of $m \times n$ matrix $A$ is orthogonal to every vector in the row space of $A$. [on hold]

How do I show? do I show this by using inner products? What should I define my inner product?
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26 views

Matrix polynomial [on hold]

Suppose: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a ...
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1answer
21 views

Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
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4answers
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Equivalent definitions of an orthogonal matrix.

I wish to show that the following definitions of an $n \times n$ real matrix $Q$ are equivalent: $QQ^T=I$, $Qx\cdot Qx=x\cdot x$ for all $x\in \mathbb{R}^n$. I found it easy to show that ...
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E as an expectation of a quadratic form [on hold]

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
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18 views

Show that every Jordan matrix has a cyclic vector

Is my following reasoning correct? Since an $n\times n$ Jordan matrix has rank $n-1$ (because we can only make the last row the zero row), its geometric multiplicity is 1, which means the matrix has ...
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1answer
38 views

Affinity of lorentz transformations

Lorentz transformations are often defined to be linear. But suppose instead we only consider transformations that preserve the spacetime interval. Is it possible to prove that those transformations ...
4
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1answer
32 views

Determinant inequality and positive definite matrix

Let $B$ and $C$ be $n\times n$ hermitian matrices, with $B$ positive definite and $C$ positive semi-definite. (1) Show that $B+C$ is positive definite (2) Show that $\det(B)\leq \det(B+C)$. What ...
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12 views

Let T be the bounded operator and T* be the adjoint operator of T.Show the following. [on hold]

Let T be the bounded operator and T* be the adjoint operator of T. Show the following. 1.||TT||=||T|| 2.||TT||=||TT*||=||T||^2 3.(T+S)=T+S* 4.(αT)=α ̄T (α∈C) 5.(TS)=S T* 6.(T* )* =T
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17 views

System of Equations for 3-digit number [on hold]

This is a rare word problem where I've had trouble: Find system of equations and use elimination. The sum of the digits of a three-digit number is 9, and the tens digit of the number is twice the ...
4
votes
2answers
49 views

Which $n$-forms are pullbacks of top forms on $\Bbb R^n$

Let $V$ be a finite-dimensional vector space. I write $F_n(V)$ for the $n$th exterior power of the dual vector space. Which elements of $F_n(V)$ can be pulled back from a top form along a linear ...