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Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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7 views

Applying a Sigma-Point Kalman Filter to State of Charge Estimation

I've found a research paper that would allow me to implement a Kalman filter to estimate state of charge for a LiPo battery, but cannot make sense of some of the symbols as I'm a first year. Can ...
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0answers
11 views

Elementary transformations of the set of vectors with respect to which the linear span is invariant

Let $L$ denote a linear space. Let $x_1,x_2,\cdots,x_n \in L$. Observe that $span\{x_1,x_2,\cdots,x_n\} = span\{x_2,x_1,\cdots,x_n\}$ This motivates the following question: under what operations ...
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0answers
26 views

How to determine the range of unknown values ​to make the matrix positive semi-determined?

$$ \left[ \begin{matrix} a & -1 & -2 & -2 \\ -1 & b & -2 & -2 \\ -2 & -2 & c & -1 \\ -2 & -2 & -1 & d \end{matrix} \right] \tag{1} $$ ...
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1answer
30 views

Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true?

Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true? $(A)$ trace of $A^2$ is positive $(B)$ $A$ has non zero eigenvalue. $(C)$ All entries of $A^2$ can't be ...
4
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1answer
36 views

Is there any inequality between 2-norm condition number and Frobenius norm condition number for rectangular matrix?

What I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix $X \in \mathbb{C}^{n \times m}$, $m=n$, then we can have, $$n - ...
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2answers
42 views

Prove that $A^2 + B^2 = O_2$ given conditions

Show that, given $A,B$ second order matrices with real entries, such that $AB = BA$, $\det{(A + iB)} = 0$ and $4 \det{A} >( \text{tr}{A} )^2$, then $A^2 + B^2 = O_2$. My progress: Considering the ...
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0answers
24 views

Multidimensional Quartic Equations

I know for the quadratic case (with $A$ an operator): $$ax^2 \Rightarrow x^T A x \Rightarrow \int xA[x]dx$$ Does any such analogy exist with $ax^4$ type functions? Either in the finite or infinite ...
4
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1answer
51 views

Different definitions of an algebra over a commutative ring

Let $R$ be a commutative ring. Here are two definitions of an $R$-algebra: An $R$-algebra is a ring $A$ together with a ring homomorphism $f: R\to A$ (Atiyah) An $R$-algebra is an $R$-module $A$ ...
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0answers
33 views

Functorial proof of Cayley-Hamilton using exterior powers

Let $V$ be a rank $n$ free module over a commutative ring $R$. Let $\dagger$ denote the adjoint with respect to the natural perfect pairing given by the wedge product $$\Lambda ^k\otimes \Lambda ^{n-k}...
3
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1answer
34 views

Show that this set is a basis for $S_1+S_2$

Let $$B_1 = \{v_1, \dots v_n, x_1 \dots x_r\}$$ $$B_2=\{v_1 \dots v_n, y_1 \dots y_s\}$$ $$B_3 = \{v_1 \dots v_n\}$$ be basis for subspaces $S_1$ , $S_2$ and $S_1 \cap S_2$ respectively. Show that ...
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3answers
29 views

Prove that the range is a subspace

In Linear Algebra Done Right, it said If $T \in \mathcal{L}(V,W)$, then range $T$ is a subspace of $W$. Proof: Suppose $T \in \mathcal{L}(V,W)$. Then $T(0) = 0$, which implies that $0 \in \text{...
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2answers
47 views

Is it possible to recognize when an endomorphism of a finite dimensional vector space is unitary for some choice of inner product?

Let $V$ a finite dimensional vector space over $\mathbb{C}$. Let $T\in GL(V)$. Are there reasonable criteria for recognizing whether or not there is some inner product on $V$ w.r.t. to which $T$ is ...
2
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1answer
52 views

How to fit data to a piecewise function?

My question today regards a set of data that I wish to fit a piecewise-defined continuous function. This data set covers a domain of x-values from $0$ to $\mu$ on the x-axis. What I need is to ...
1
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2answers
41 views

all invariant subspaces

Let $A$ be a linear operator such that $A \begin{bmatrix}x\\y\\z \end{bmatrix}=\begin {bmatrix}x+y\\y-x\\0 \end{bmatrix}$. Find all invariant subspaces of $A$. I know how to find subspaces of ...
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0answers
21 views

What is the equation for an ellipse in standard form after an arbitrary matrix transformation?

Suppose I have a general ellipse parameterized by a center point $(h, k)$, semimajor axis $a$, semiminor axis $b$, and rotation angle $\theta$. This has the formula in standard form of $\frac{((x−h)\...
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0answers
61 views

Is there a set of three linearly independent vectors in $\mathbb{C}^2$?

I claim the set of three linearly independent vectors in $\mathbb{C}^2$ to be empty. But I'm not sure if the following proof is correct. Let $x$ and $y$ be linearly independent vectors in $\mathbb{C}...
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2answers
46 views

No nontrivial subspaces implies irreducibility of characteristic polynomial

Let $k$ be a field and let $V$ be a $k$-vector space. Let $T:V\to V$ be a linear transformation, and suppose $T$ has no invariant subspaces in $V$ other than $0$ and $V$ itself. Does it follow that ...
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0answers
40 views

Find the vector equation for a line that passes through $P$ and intersects $L$

I just need to know if my answer is right. If it isn't please tell me what the answer is and what I did wrong. Question: Let $L$ be the line with parametric equations \begin{align*} x & = 5−...
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4answers
35 views

Show that $(\lambda v= \vec{0})\Rightarrow \lambda=0$ or $v=\vec{0}$ Vector-space

If $\lambda \neq 0$ then $v= 1v=(\lambda\lambda^{-1})v=\lambda v \lambda^{-1}v=\vec{0}\lambda^{-1}v=\vec{0}$ Since $\vec{0}x=\vec{0},\forall x\in V$ But what if $v\neq \vec{0}$
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1answer
28 views

How to determine eigenvectors of symmetric circulant matrix {{A,B,B},{B,A,B},{B,B,A}}?

I'm trying to find the eigenvectors for the matrix $$\begin{bmatrix} A & B & B \\ B & A & B \\ B & B & A \end{bmatrix} $$ . I determined the eigenvalues to be $\lambda_1=\...
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2answers
37 views

Can $\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$ be solved for $\lambda$?

In this post I suggested that the expression $$ \| \left( X'X + \lambda I \right) ^{-1} X'y \| = t $$ couldn't be easily solved for $\lambda$, because you need to "invert" the norm. But in general ...
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0answers
22 views

What kind of matrices can be visualized by graphing the column vectors?

In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis ...
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1answer
24 views

Determine the dimension and find a basis of a vector space

$V_1 = (x_1, ..., x_7)^T \in \mathbb{Z}^7_{5} : x_1 + 3x_2 +x_3+2x_4+3x_5+x_6+2x_7 = 0$ $3x_1 + 4x_2 +3x_3+x_4+4x_5+2x_6+4x_7 = 0$ $2x_1 + x_2 +4x_3+4x_5+x_6+2x_7 = 0$ I am supposed to find ...
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1answer
19 views
0
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1answer
17 views

Matrix multiplication, linear transformations and systems of equations

In linear algebra, I learned that the reason why matrix multiplication is defined in its way is because of composition of linear transformations. So, multiplication of two matrices refers to the ...
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2answers
19 views

Let $v \in R^{k}$ ,with $v^{T} v \neq 0$ . Let $P=I-2 \frac{v v^{T}}{v^{T}v}$, where $I$ is the $k X k $ identity matrix.

Let $v \in R^{k}$ ,with $v^{T} v \neq 0$ . Let $P=I-2 \frac{v v^{T}}{v^{T}v}$, where $I$ is the $k \times k $ identity matrix. Then prove that eigenvalue of $P$ are $1,-1$ and $P^{2}=I$ $P^{2}=(I-2 \...
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2answers
44 views

Matrix equations, simplify them.

I have some equations and I don't know am I doing the simplification right. Can someone check it? For example, we have an equation $AX = 4X + B$, where $A$ and $B$ are matrices. So, what I have done ...
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1answer
25 views

Chromatic polynomial properties

I have some properties which I do not understand how to form a proof for. Most of them are by induction, which is not my strong point Any help would be fantastic. Looking at the chromatic polynomial ...
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2answers
49 views

Proof in linear algebra/calculus

So I am currently studying Calculus and Linear Algebra and I came across the same concepts that is being applied in a lot of the proofs that I read for Calculus and Linear algebra but not capable of ...
3
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0answers
55 views

Rewrite a system of linear equations

Fix $r$ and $d\equiv r+\binom{r}{2}$ and consider $d$ real numbers $b_1,b_2,..., b_d$. I have the following system of linear equations: $$ \begin{cases} \text{$(\diamond)$ Differences between the ...
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1answer
22 views

Inverting variable size block matrix

Given a block matrix $M$ $$\bf M=\left(\begin{array}{cccc} a\ \mathbb{I}_{2} & \boldsymbol{\boldsymbol{A}}_{12} & \boldsymbol{A}_{13} & \boldsymbol{0_{(2,3)}}\\ \\ \boldsymbol{A_{21}} &...
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1answer
23 views

Is $C(A)+C(B)=C(A+B)$, if $C(A)\cap C(B)= \{0\}\,$?

Is $C(A)+C(B)=C(A+B)$, if $C(A)\cap C(B)= \{0\}?$ where $C(A)$ denotes the column space of $A$. It's true that, $\DeclareMathOperator{\rank}{rank}\rank(A+B)=\rank(A)+\rank(B),$ if $C(A)\cap C(B)= \{0\...
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2answers
27 views

Prove every finite field is of cardinality $p^n$ [duplicate]

I realise that this question has been asked multiple times before, but I would like to ask about a specific detail in my proof. Let F be a finite field. I begin by showing that the characteristic of ...
1
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1answer
50 views

The set of real structures on $\mathbb{C}^n$ is isomorphic to $GL(n,\mathbb{C})/GL(n,\mathbb{R})$ as a $GL(n,\mathbb{C})$-space.

In representation theory a real structure on a $G$-module $V$ (finite dimensional complex vector space, in which the group $G$ has a linear action on it) can be definide as a conjugate-linear $G$-map $...
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0answers
15 views

Help to check first and second order derivatives

I have the following function: $$y=\left [ \alpha \eta k^{-c(1+\rho )}+\beta k^{-\rho} \right ]^{-\frac{1}{\rho}}$$ Of which I have to calculate first and second order derivatives. First order ...
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1answer
23 views

scale a matrix to unit sum [on hold]

I have a matrix lets say a 3x3 H = [ 0.2 0.8 -1 -0.3 0.5 0.2 0.1 0.1 -0.2] I want to: a) scale this matrix such that the absolute value of the ...
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0answers
23 views

showing that Symplectic matrix is diagonalizable [on hold]

please how to Show that any symplectic matrix is ​​diagonalizable by using the fact that every unit matrix is ​​diagonalisable.
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0answers
6 views

On $QR$ factorization of upper Hessenberg matrices with real entries

Let $A$ be an $n\times n$ invertible upper Hessenberg matrix (https://en.wikipedia.org/wiki/Hessenberg_matrix) with real entries. I am trying to prove the following three statements : (1) There ...
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1answer
29 views

Showing that a matrix F is a matrix representation of $\mathbb{R}$

The question is given below and its answer are given below: 1-I know that according to Vinberg book which is called "Linear representations of groups" all finite dimensional differentiable ...
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2answers
40 views

$A^{2n}=I$ but $A^{n}\neq I, -I$ [duplicate]

Let $A$ be a $n\times n$ real matrix. such that $A^{2n}=I$ but $A^{n}\neq I, -I$, $n\geq 2?$ I have a example for $A^2=I$ but $A\neq I, -I$ but could not find a similar example for this question. I ...
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0answers
18 views

Proving the Linearity of Expectation from the opposite side

An alternative method of variance is $$E[X^2]-(E[X])^2$$ and proving that $E[(X-E[X])^2] = E[X^2]-(E[X])^2$ is done. But how about the other way around $E[X^2]-(E[X])^2 = E[(X-E[X])^2]$, how can you ...
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0answers
31 views

Is $A$ is diagonalizable/non-diagonalizable? [on hold]

Let $A$ $\in $$\mathbb{M}_5({\mathbb{C}})$ matrix such that $(A^2-I)^2=0$. Assume that $A$ is not a diagonal matrix. Which of the following statement is true ? $1)$$A$ is diagonalizable. $2)$ $A$ ...
0
votes
1answer
21 views

Partial trace and linear operators

Trying to solve the following: Consider a linear map as follows $$vec: L(X,Y)\rightarrow Y \otimes X $$ $$ vec: E_{b,a} \mapsto e_b \otimes e_{a}$$ which can be looked as a change of a basis map. ...
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1answer
30 views

Please help me Calculate the minimal polynomial.

Please help me out to calculate the minimal polynomial of this matrix A where; $$ A=\begin{pmatrix} 0 & 0 & 0 & -4\\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 5 \\ ...
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0answers
9 views

On positive definiteness of a sub-matrix after first step Gaussian elimination to a symmetric positive definite matrix

Let $A=[a_{ij}]\in M_n(\mathbb R)$ be a symmetric positive definite matrix (i.e. all eigenvalues of $A$ are real and positive ) with $a_{11}\ne 0$ . Now let $A_1=[a' _{jk}] \in M_{n-1}(\mathbb R)$ ...
0
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2answers
30 views

How can I prove that?

If T : V → W is a linear transformation. If $\ker T$ has $\{u_1,\dots,u_n\}$ as basis. If image of $T$ has $\{T(v_1),\dots,T(v_m)\}$ as basis. How can I prove that $$ \{v_1,\...
1
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0answers
18 views

An affine transformation, and its effect on a curve and a polynomial.

Suppose $(u,v) = A(x,y)$ is affine transformation. Where $u = ax + by + e$, and $v = cx + dy + f$ , and the inverse transformation given by $x = a'u + b'v + e'$ and $y = c'u + d'v + f'$. Suppose ...
2
votes
1answer
27 views

Sum of vectors in any linear space

Suppose that vectors $x_1,x_2,…,x_n$ have the following property: for each $i$ the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1,x_2,\dots ,x_n$ are not ...
0
votes
1answer
20 views

The rank of a matrix of Gaussian random vectors

If I generate a matrix $R\in\mathbb{R^{k\times p}}$, $k<p$, with i.i.d. entries $R_{i,j}\sim\mathcal{N}(0,\sigma^2)$, is there a guarantee that this matrix will have rank $k$, i.e. its columns will ...