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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Computing the inverse of a unimodular matrix

Is there a specialized/simpler way to compute the inverse of a unimodular matrix? I am looking for something simpler than computing the adjoint matrix.
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What is $\frac{d\|\mathcal{A}(X^TX)-b\|_2^2}{dX}$?

Let $X \in \mathbb{R}^{m\times n}$, $b \in \mathbb{R}^{1\times s}$ and $\mathcal{A}$ a linear operator from $\mathbb{R}^{n \times n}$ to $\mathbb{R}^{1 \times s}$. How do I find ...
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15 views

Recurrence relation-Summation of a series [on hold]

Sir, I have a Converging recurrence relation given as below, $-(\psi(n-1)) ...
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463 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
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54 views

How to find a basis for a linear space of polynomials?

Questions such like, (1) Let $V = P^4$ be the vector space of all real valued polynomials of degree less than or equal to four. Let $W =\{p(x)\in P^3 |p(−2)=p(2)\}$. Find the basis for $W$ (2) Let ...
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27 views

Lipschitz continuity of inverse

Given a function f : $\mathbb{R}^n\to\mathbb{R}^m$, which is known to be Lipschitz continuous, can we say anything about the Lipschitz continuity of it's inverse function (in this case, the ...
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66 views

Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
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1answer
60 views

Which eigenvalues and eigenvectors of a linear system are of interest and what to do with them? [closed]

Given a linear system of equations, is there any "general" criteria how to select eigenvalues and eigenvectors after computing them? For example, positive or negative, the biggest or the smallest, ...
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47 views

System of equations with a unique solution, no solution or an infinite number of solutions

I was doing a past OCR Further Pure 1 Paper from January 2011, but came across the following question that I could not solve, even with the help of the mark scheme: Determine whether the ...
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2answers
61 views

Dimension of $\dim_{\mathbb C}\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)$

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}\mathbb C[X,Y]/I$. Obviously ...
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39 views

Sesquilinear forms seen as bilinear maps

Let $V$ be a complex vector space. A sesquilinear map (or conjugate-linear in the first variable and linear in the second) on a complex vector space $V$ is a map $f: V \times V \rightarrow \mathbb{C}$ ...
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29 views

Transformation of a sum of dot products

I'm not quite sure about this, So I'd like if someone could help me. Can someone explain to me how they get from 2 to 3? \begin{align}\left<2u,u+v\right> &= 0\tag1\\ ...
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1answer
31 views

Explanation of the Gram-Schmidt orthogonalization process

There is a proof of Gram–Schmidt orthogonalization in Kolmogorov's book. Can you explain $h_n$ and how do we write $f_n=b_n\varphi_1+\cdots+b_{n,n-1}\varphi_{n-1}+h_n$? My main question is why does ...
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1answer
216 views

Reduced Row Echelon form in excel

I was wondering if there was a way I could enter some matrix in excel like ... 1 -1 -1 0 15 5 0 -5 0 -5 10 8 With each number in a cell, and then have excel transform this matrix into reduced ...
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1answer
24 views

Derivation of Mahalanobis Distance

I was recently reading up on the Mahalanobis Distance, and understood how it generalizes distance measures for multivariate data such as the Euclidean Distance. However, what got me wondering was how ...
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31 views

When a system of rational linear homogeneous equations have complex solutions

Question: When a finite system of rational linear homogeneous equations in finitely many variables have a nontrivial complex solution (that is not a rational solution), does it imply that there is ...
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21 views

Finding this Polynomial Subspace

Let $A = k[x^{\pm 1}, y^{\pm 1} ] $, considered as a $k$ - algebra. Can someone give me a nice description of the (vector) subspace: $$ A_0 = \lbrace (f,g) \in A^2 : \frac{ \partial f}{\partial y} = ...
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138 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
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30 views

The determinant of adjugate matrix

Why does $\det(\text{adj}(A)) = 0$ if $\det(A) = 0$? (without using the formula $\det(\text{adj}(A)) = \det(A)^{n-1}.)$
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Diagonalization of $M=ab^t+ba^t$

Given $a=(a_i)_{i=1}^n$ and $b=(b_i)_{i=1}^n$ in $\mathbb{R}^n$, we define the matrix $M=(m_{ij})_{i,j=1}^n$ as: $$ m_{ij}=a_ib_j + a_jb_i, $$ or equivalently $$M=ab^t+ba^t.$$ What are the eigenvalues ...
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17 views

Translation of basis for a vector space on the specified distance

In the Euclidean space $XYZ$ is a basis $X_1Y_1Z_1$ defined that is specified by the vectors $\overrightarrow {O_1X_1}$, $\overrightarrow {O_1Y_1}$ and $\overrightarrow {O_1Z_1}$. How to calculate ...
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1answer
236 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
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co and contravariant vectors, their difference and properties

Very often when talking about covectors, co- and contravariant stuff, it's mentioned that there is no difference in "normal" linear algebra. That the difference only comes "when dealing with curved ...
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331 views

If $A^2=2A$, then $A$ is diagonalizable.

My brain has already burned. I think, I should use a double linear transformation but can't find any proper solution. Let $\mathbb F$ be a field and $A\in M_n (\mathbb F)$ satisfying the equation $$ ...
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2answers
245 views

$AB=BA$ implies $AB^T=B^TA$.

I am looking for an elementary proof (if such exists) of the following: $$ AB=BA \quad\Longrightarrow\quad AB^T=B^TA, $$ where $A$ and $B$ are $n\times n$ real matrices, and $A$ is a normal matrix, ...
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458 views

Prove that $\det(A+B)=\det B$

Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy $$ A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad AB=BA. $$ Prove that $$\det(A+B)=\det B.$$
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1answer
26 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
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2answers
213 views

Relation between singular values of a data matrix and the eigenvalues of its covariance matrix

I have a $m \times n$ data matrix $X$, ($m$ instances and $n$ features) on which I calculate the Covariance matrix $$C := \frac{1}{(m-1)} X'X.$$ Then I perform eigenvalue decomposition of $C$, and ...
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74 views

relative sign in Hodge star of tensor product

Let $V$ be a vector space of arbitrary (finite) dimension and let $(V, \langle \ ,\ \rangle, I) = (W_1, \langle\ ,\ \rangle_1, I_1) \oplus (W_2, \langle\ ,\ \rangle_2, I_2)$ be a direct sum ...
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1answer
19 views

Corollary to Smith normal form

Given the following theorem (Smith normal form) Let $R$ be a PID and A a $(m \times n)$-matrix over $R$. Then there exist square invertible $R$-matrices $P$ and $Q$ such that $A'=PAQ$ is a diagonal ...
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30 views

Change of basis from falling powers to powers for polynomials up to degree $n$

Notice that $$(1, x, x^{\underline{2}}, x^{\underline{3}}, \dots)$$ and $$(1, x, x^2, x^3, \dots) $$ both are bases of $\mathbb{R}[x]$ (where $x^{\underline{n}}$ is the falling power). Now suppose the ...
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Bilinear form equations [on hold]

$\mathit{V}$ is 3-dimensional vector space over complex. Complex valued square matrix $R_A, R_B, R_C\in M_3(\mathbb{C})$ are given. Suppose that both $R_A, R_B$ and $R_C$ are Hermitian ...
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1answer
33 views

Norms are continuous maps [on hold]

How can I prove that the mapping $$ f:\mathbb{R}^{N}\rightarrow\mathbb{R}, $$ defined by $$ f(\mathbf{x})=\|\Phi\mathbf{x}\|^{2}_{2}, $$ is continuous?
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179 views

If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$

$A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$ If $\lambda$ is an eigenvalue ...
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477 views

What free software can I use to solve a system of linear equations containing an unknown?

Question: What free software can I use to solve a system of linear equations $M\mathbf{x}=\mathbf{y}$ where the entries of $\mathbf{y}$ vary with an unknown quantity $n$? Presumably I could do ...
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3answers
95 views

Solving a system of three linear equations with three unknowns

Is my working correct or am I completely wrong? Have I missed anything out? Any feedback is appreciated. Question: Consider the following system of equations $2x + 2y + z = 2$ $−x + 2y − z = −5$ ...
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1answer
15 views

Weighted average of multiple points

Let's say I have a triangle whose three corners are $$(x_1,y_1),(x_2,y_2),(x_3,y_3).$$ I have a weight assigned to each one as a percentage, so the first point might be $75\%$, the second $15\%$ and ...
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1answer
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Question about intersections [on hold]

"Consider the intersection of the functions $y=\frac{m}{10x} + m$ and $y=\frac{m}{x}$ where $m$ is a real number. Investigate the values of m that may provide, 0, 1, or 2 points of intersection." ...
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201 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
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1answer
23 views

How could I calculate the local size of an object given its distance and actual size?

Lets say I take a picture of a sign. I know that sign is 20ft (width), 10ft height. I'm standing 40 feet away. If I were to take a picture, how could I calculate how wide and how high the sign is in ...
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1answer
44 views

Eliminate 2 variables from 3 equations with lots of parameters

I want to eliminate the variables x and y from these 3 equations in a way that all parameters appear in one equation without x and y: ...
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46 views

a proof question regarding to eigenvalues and diagonalization [on hold]

Let the scalar field be $\mathbb{F}$. Let $T: V\rightarrow V$ be a linear operator represented by the $n\times n$ matrix $A= [T]_{\alpha\alpha}$. Suppose that the characteristic polynomial of $A$ ...
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1answer
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Simple proof that a $3\times 3$-matrix with entries $s$ or $s+1$ cannot have determinant $\pm 1$, if $s>1$.

Let $s>1$ and $A$ be a $3\times 3$ matrix with entries $s$ or $s+1$. Then $\det(A)\ne \pm 1$. The determinant has the form $as+b$ with integers $a$,$b$ and it has to be proven that $a>0$ if ...
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1answer
67 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
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1answer
31 views

Projection equation

I'm a programmer, not a math expert or statistician by any means, but my organization wants a page in our admin console that displays a projection of how many registrations we can expect to see based ...
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1answer
30 views

Determine whether the following map is a linear transformation.

So I have to determine if the following is a linear transformation: $$T: F(I) \rightarrow F(I)$$ defined by: $$T(f) = 2f$$ I know that if you let $T: V\rightarrow W$ be a linear transformation. Then: ...
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Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
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1answer
67 views

Is it true that $u + v$ is an eigenvector corresponding to the eigenvalue $\lambda$?

Let $A$ be an $n \times n$ matrix, and $u, v$ be eigenvectors corresponding to an eigenvalue $\lambda$ of $ A$ (that is, $Au = \lambda u$ and $Av = \lambda v$). Is it true that $u + v$ is an ...
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1answer
31 views

Isometry in a finite dimensional vector space is always surjective

My book defines an isometry as a linear operator between two vector spaces X and Y where: $$\|T(x)\|=\|x\|$$ Later it has a sentence which I do not understand. If we have a finite dimensional ...
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2answers
22 views

Largest eigenvalue of a block diagonal matrix is an eigenvalue of the largest block?

Consider this square matrix $C = \begin{bmatrix} A& 0 \\ 0 &B \end{bmatrix}$, where $A$ and $B$ are also square matrices. Suppose $A$ is larger in the sense that is an $n \times n$ matrix, ...