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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Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
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matrix differentiation - derivative of matrix vector dot product with respect to matrix

Given the function $$f(N) = x_1^T M x_2 $$ where $x_1 = Nv_1 $ $x_2 = Nv_2 $ $x_1, x_2, v_1, v_2$ are vectors with dimension $n \times 1$ $M$ and $N$ are matrices with dimension $n \times n$ ...
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Doubt about subspaces being vector space

Whenever i am saying $V$ is a $n$-dimensional vector space, it means it has $n$ basis vectors each with n elements, right. So when i am proving some theorems or relations involving some ...
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How to convexify (relax) this L0 eigenvalue optimization problem?

Let $C_1,\dots,C_L$ be $N\times N$ hermitian matrices. Let $d<0$ be a given negative constant. Then consider the optimization problem \begin{align} \max_{r\in \mathcal{R}^{L\times 1}} &\mid\mid ...
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Let $u_1 = (a,1,-1)$, $u_2 = (-1,a,1)$, $u_3 = (1,-1,a)$. For what values of a are $u_1, u_2, u_3$ linearly independent?

Let $u_1 = (a,1,-1)$, $u_2 = (-1,a,1)$, $u_3 = (1,-1,a)$. For what values of a are $u_1, u_2, u_3$ linearly independent? Can someone show me the way to solve this kind of question? Thank you.
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The i's and j's in a Matrix

I know that i means row and j means column, what i don't understand is what are they meaning when they say that the row is greater than or = to 1? And the column is less than or equal to 3? I don't ...
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68 views

Finding eigenvalues of a block matrix

I have a block matrix of size $2N \times 2N$ of the form $$B = \begin{bmatrix} A_N & C_N \\ C_N & A_N \end{bmatrix}$$ where $A_N$ and $C_N$ are both $N \times N$ matrices. Specifically, $$A_N ...
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Can an Elementary Matrix's Inverse's Determinant = 0?

Can someone explain to me why an elementary matrix's inverse determinant cannot equal 0? Or can it? Is there some theorem to elementary matrix inverses? THANKS FOR YOUR INSIGHT! :)
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Inverse of LU decomposition

The LU decomposition is $A=LU$, where $L$ is lower and $U$ is upper triangular. For the example of a 3*3 matrix $$A= \begin{pmatrix} 1 & 0 & 0 \\ l21 & 1 & 0 \\ ...
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Given the position and number of columns find the coordinates of the array

Given the following matrix. If the position in the array can be found using the formula pos = y * number_of_columns + x; given x, y and number_of_columns. ...
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Convergence of Baker-Cambpbell-Hausdorf for compact groups

It is well known that the Baker-Campbell-Hausdorf formula doesn't need to converge for general elements of a Lie algebra, resp. for matrices with norms larger then 1. On the other side, if $G$ is a ...
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Transpose : Matrices and Orders of

Here is my Answers and Reasoning, all I ask is that you check it and direct me if I have gone wrong, Thank you! Number 2 - because (BC)^T equals C^T x B^T which is a 5x2 matrix. This multiplied by ...
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Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$? ...
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Prove this inequality using inner product definition.

Prove that $(x_1 + \dots + x_n)^2 \leq n({x_1}^2 + \dots + {x_n}^2)$ for all positive integers n and all real numbers $x_1, \dots x_n.$ What I tried: Taking the square root of both sides you get ...
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Help for understanding Danielson-Lanczos lemma

The Danielson-Lanczos lemma is the basis for fast Fourier transform algorithms. Now, I do understand this step $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N} = \sum_{n=0}^{(N/2)-1} ...
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A question about the representation of coordinates with respect to different bases.

Let $B$ and $B'$ be separate basis matrices for a vector space $V$. Each column is a separate basis element. Let $X$ be $X'$ be the coordinates for the same vector $v\in V$ in terms of the bases $B$ ...
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Practical Application of Linear Transformation as taught in Linear Algebra.

Can anyone provide a basic practical use of linear transformation? ( Or maybe a metaphor like example) Also visual, practical, or real applications for one to one onto null(A) determinant rank ...
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35 views

Approximate Equivalent To Michael Spivak's text, “Calculus” but for Linear Algebra?

Does anyone know of an approximate equivalent To Michael Spivak's text, "Calculus" but for Linear Algebra? I love the way this book is written! It is simultaneously rigorous and thorough without ...
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To find matrix from given linear transformation

Let $\vec{x}$ & $\vec{y}$ be linearly independent vectors in $\mathbb{R}^2$. Suppose T: $\mathbb{R}^2$ $\rightarrow$ $\mathbb{R}^2$ is a linear transformation such that $T \vec{y}$=$\alpha ...
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Eigenvectors and the relationship between variables in a system of equations.

I am learning about complex eigenvalues in Linear Algebra and I am confused with one problem. I have a matrix in $A-\lambda I $ form. For the eigenvalue $\lambda=3+2i$, $A-\lambda I=\begin{bmatrix} ...
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Help to understand and verify this question regards Spectral Thereom

But I don't how to verify that it is an orthonormal basis, because their inner product doesn't equal to 0
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Diff. Eq. Example with Matrices

I'm currently working on a side project of mine that deals with $\sin(A)$ and $\cos(B)$, where $A,B\in\mathbb{C}^{nxn}$. I'm trying to find some interesting (or non-interesting) examples where one ...
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Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
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What shall the characteristic of K be for the identity (1) to be valid?

Let K be a field. Let f be a linear operator on V. Consider the identity: v=1/2(v+f(v))+1/2(v-f(v)) Suppose, f is such that for any v in V, one has: f(f(v))=v (1) What shall the characteristic of K ...
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Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
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Orthogonality of Sub Sections of Orthogonal Vectors

Given a sent of $ N $ orthogonal vectors $ {\left \{ {v}_{i} \right \}}_{i = 1}^{N}, \ {v}_{i} \in R^{N} $ Now, let's say K samples of the same index (Lets say j = 51:100, where N = 200) are removed ...
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35 views

Clarification for a proposition about inner products

Proposition. If $V$ is a complex inner-product space and $T$ is a linear operator on $V$ such that $$\langle Tv, v\rangle = 0$$ for all $v \in V$, then $T = 0$. $$\begin{align*} \langle Tu, ...
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Need help understanding the proof: if v is a left singular vector of A then v is a unit eigenvector of $AA^{T}$

This is the proof in my textbook: What I don't understand it why " $AA^{T}u = 0u $ means that u is an eigenvector. Is this a theorem that I don't know? That if you multiply a matrix by a vector ...
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Find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace spanned by $g(x)=x−12$ and $h(x)=1$.

Use the inner product $\langle f,g\rangle =\int_0^1 f(x)g(x)dx$ in the vector space $C^0[0,1]$ to find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace $V$ spanned by $g(x)=x−1/2$ and ...
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32 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
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Showing a basis exists for a particular transformation

Let $V$ be $n$-dimensional vector space over reals. $T: V \to V $ linear of rank $r < n$ , $T^2 = T$. Show $V$ has a basis ${v_1...v_n}$ s.t $T(v_i) =v_i$ for $i=1...r, 0$ otherwise. Could someone ...
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Proving isomorphism between a quotient space of continuous functions.

This is really a follow-up question to this question, in the sense that it arose from that question. You don't need to read that question for this to make sense. To be proven: Let $V=\mathcal ...
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How to find a nonlinear function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is almost linear in the sense $f(\alpha (a,b))=\alpha f(a,b)$? [duplicate]

I need to find a nonlinear function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that $f(\alpha (a,b))=\alpha f(a,b)$ for all $(a,b)\in\mathbb{R}^2$ and $\alpha\in\mathbb{R}$. I can't find anything. ...
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Linear independence of vectors over a subspace

I cannot understand the difference between these two statements: vectors $g_1,...,g_k$ are linearly independent over the subspace $L\subset K$. vectors $g_1,...,g_k$ are linearly independent in the ...
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Intersection of lines in 3D space [closed]

Given two or more pairs of points in 3D space, I should calculate the intersection of the lines passing through each of these pairs of points: for each pair of points, I get the linear system of two ...
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Two matrices have same eigenvalues.

Setup Let $n \geq 2$ and consider the matrices $A_n, B_n$ defined by $A_n = (a_{ij})$, $B_n = (b_{ij})$, where $a_{ij} = 1$ when $j \equiv 1 \pmod{2}$ and $i = j + (j+1)/2$ or $j \equiv 0 \pmod{2}$ ...
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1answer
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Find a linear operator given the kernel

"Find a linear operator $T:\mathbb{R}^3\to\mathbb{R}^3$ so that the kernel is generated by $(1,2,-1)$ and $(1,-1,0)$." It's been a while since I've worked with linear algebra, but from memory I know ...
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1answer
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Verification Matrices & Linear Equations Part 2

...Continued Question 3 A - True because if it equals 4 then there will be infinite solutions B - True because any gradient except for one that is equal (4) will intersect giving a unique ...
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Strictly Convex Functions

I am trying to show the equivalence of two definitions of strictly convex functions. Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth function. The function $f$ is strictly convex if for each ...
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Prove that solution set of interval linear equations is continuous over projective space

I first have to give some quick theory. I am working with Interval Linear Equations. A superscript L (or R) denotes the left (or right) endpoint of an interval. Thus if $X = [a,b]$, then $X^L = a$ ...
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JCF given characteristic polynomial

A $15 \times 15$ matrix $M$ with complex entries has characteristic polynomial equal to $(x-1)^7(x-2)^8$. Find all possible minimal polynomials for $M$ such that the characteristic and minimal ...
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Nonhomogeneous Systems of m equations in n unknowns and Solution Spaces.

My book says that solutions sets of nonhomogeneous systems of m equations in n unknowns is NEVER a subspace of R^n. Why? If we look at any two planes intersecting in R3, there may be a line formed. ...
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Interpreting complex coefficients in the Ritz procedure

I am using the Ritz procedure to write a trial function as the superposition of other admissible functions, with the coefficients being unknown variational parameters to be determined. The variational ...
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Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
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Vector Orthogonality and Length

The problem statement, all given variables and data Let $\textbf{a}$, $\textbf{b}$ be two vectors in $\mathbb{R}^n$. If $\textbf{a} + \textbf{b}$ and $\textbf{a} - \textbf{b}$ are orthogonal, then ...
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decomposition of rectangle matrices over any field

Let $A\in M_{n\times m}(\mathbb F)$ be a matrix, and $\mathbb F$ is a field. My question is does there any interesting decomposition of $A$ to a product of matrices. We know a lot of decompositions ...
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Acute angle between plane and line

Find the acute angle between: $x-y-3z=5$ and $x=2-t$ $y=2t$ $z=3t-1$ Here is how I proceed. I take the dot-product of the normal of the plane and the directional vector of the line. This gives me ...
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Doubt in proof of Dual of the direct sum

If $M$ and $N$ are subspaces of $V$, and if $V = M \oplus N$, then $$V' = M^\perp \oplus N^\perp$$ where $W^\perp$ is the annihilator of $W$. I didn't understand how to prove both of the ...
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What conclusions we can draw from $|A|$=0? if $A$ is a positive-semidefinite matrix [on hold]

What conclusions we can draw from $|A|$=0? (the determinant is 0) $A$ is a positive-semidefinite matrix. Many thanks