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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How to applied Gaussian Elimination for non-full rank matrix

I have a question about gaussian elimination. I want to find solution of $$Ax=b$$ as soon as possible using Gaussian Elimination. This is my matrix A ...
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Example of matrices with some interesting properties like same characteristic and minimal polynomial etc.

Looking for two matrices $A$ and $B$ with entries in the field $F_2$ with the following properties: $A$ and $B$ both are invertible,have same minimal polynomial,Characteristic polynomial,same ...
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19 views

Some Dense subset of $M_2(\mathbb{R})$ with its usual topology?

The set of all invertible matrices i.e $GL_2(\mathbb{R})$ The set of all matrcies having both real eigen values. Having $Trace(A)=0$ $3$ is not dense set as It is closed set! $1$ Is dense. take ...
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1answer
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Definition of angle between vectors in spaces with dimensions n

I am working on a problem which requires me to find certain values of the components of vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^4$ such that the angle between them is $\pi/3$ If my ...
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1answer
23 views

Analogy of transpose for a function?

In the page 2 of Linear algebra explained in four pages reference, it has a box describing the relationship between functions and linear transformation. It states that the set of zeroes of a function ...
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Does Least squares solution exist for this case?

$ {\bf{Z}} = {\bf{H}} \cdot {\bf{S}} + {\bf{N}} $ Dimensions of the matrices are as follows: Z = m X m H = m X n S = n X m (matrix S is an orthogonal matrix) N = m X m. All the elements of the ...
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Looking for proof of “ two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues ”

For any real square matrix $X$ let $P(X)$ denote the no. of its positive eigenvalues counting multiplicity . Let $A$ be a real symmetric $n \times n$ matrix and $B$ be a real invertible $n \times ...
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28 views

Complex projections order in inner product

So the complex projection is defined as $$\operatorname{proj}_\vec{u} \vec{v} = \frac{\langle \vec{v},\vec{u}\rangle}{\langle\vec{u},\vec{u}\rangle} \vec{u}$$ with complex inner product. I was ...
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22 views

What exactly is Standard Coordinates?

What exactly is a standard coordinates? Sorry it seems like a very stupid question, but my professor didn't really explain it and just started to use it for solving other problems related to ...
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29 views

establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$

establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$. I couldn't understand how to solve it please just give me the first step, maybe I can figure out the ...
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1answer
24 views

If $v\not=0$, then $v/\|v\|$ has norm 1

The question is: Show that if $\vec v$ is a non-zero vector in $\mathbb R^n$ then $\left( \dfrac{1}{||\vec v||} \right ) \vec v$ has norm $1$. I assume that $\vec v=(v_1,v_2,v_3,...,v_n)$ , ...
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1answer
20 views

Verify if symmetric matrices form a subspace

I need to verify if the symmetric matrices form a subspace. But I don't know how to represent a general symmetric matrix. I know that the matrix $A$ is symmetric if $A = A^t$ but I can't write a ...
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1answer
30 views

Obscure Rotation Matrix

The points of the shape after the shear are $(-0.25, 1)$, $(1.75, 1)$, $(0.25, -1)$, $(-1.75, -1)$. Other than that the only other information given is that the vertical edges of the original shaded ...
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24 views

$C$ & $D$ stuck at calculating

Hello! I am super stuck about how to calculate $C$ and $D$ in the example image, I know it's something simple I just can't figure it out!
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1answer
44 views

How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
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1answer
26 views

$r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
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8 views

Matrix coordinate blocks and null space of a vector

Suppose we have some non-zero $c \in \mathbb{R}^n$. Let $\{U_i\}_{i=1}^m$ be a collection of $n \times n_i$ matrices where each column is a standard basis vector. Suppose that each standard basis ...
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1answer
38 views

$A^{T}b$ inconsistent system!

I am trying to figure out how the calculation on the last image comes to be (question 9, the yellow area). I have calculated the rest without issue. I know that the formula for the last set is ...
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Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
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2answers
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If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
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A reduction of Cayley-Hamilton to the complex case [on hold]

I found an interesting proof of the general Cayley-Hamilton theorem but i didn't quite get all of it. I'm trying to make sense of this reduction argument. I'm not that familiar with algebras so this ...
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1answer
15 views

Find a basis where vector has fixed cooridnates

What's the method for finding basis where $\beta_1=(0,2,1) \ , \ \beta_2=(1,1,2)$ and $\beta_1$ has cooridnates $1,2,-1$ and $\beta_2$ has $0,0,1$ ? we have that $(1,1,2)=(a_3,b_3,c_3)$ and ...
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A linear system for which the solution space is spanned by the given vectors [duplicate]

Make a system with $3$ equations and $3$ unknowns of which the solution space $V$ is spanned by the column vectors: $$\begin{bmatrix} 1 \\0 \\-1\end{bmatrix},\quad \begin{bmatrix}1 \\3\\ ...
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What is recommended for studying Linear Algebra? [duplicate]

I don't know anything about linear algebra and want to start afresh but in a proper mathematical manner. What should I do and which are recommended? I know the basics of multi-variable calculus, some ...
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23 views

Find a family of linear inequalities for which the unit ball is a solution set

Find a family of linear inequalities for which the unit ball $A = \{x \in \mathbb R^n \ | \ \|x\|_2 \le 1\}$ is a solution set Would it just be $x_1^2 + ... + x_n^2 \le 1$?
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2answers
124 views

Algebraic operations with tensor products and direct sums.

When first learning about tensor products and direct sums of vector spaces, the analogy of multiplication and addition of vector spaces is sometimes used to help with the intuition. If we look at the ...
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Help with an Analysis problem involving isomorphisms and volumes.

I would like some help solving the following problem: Let $f: U \subseteq \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ be a class $C^{1}$ function. Suppose that for some $a \in U$ the derivative of $f$ ...
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1answer
26 views

prove linear independence of polynomials

Let $f_1,...,f_k$ be polynomials at field $K$ not including the zero polynomial and $\deg f_i \neq \deg f_j$ for every $i \neq j$ Show that polynomials $f_1,...,f_k $ are linear independent. I don't ...
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1answer
38 views

A Challenge on linear functional and bounding property

I took a midterm exam and after that wrote this problem down. My instructor was unable to solve it. The problem is copied here in order for anyone to help me. Suppose $f:E\to \mathbb{C}$ is a ...
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Find all 2 x 2 skew-symmetric matrices A [on hold]

Find all $2 \times 2$ skew-symmetric matrices $A$, if any, such that $A^2 + I_2 = 0$ Please help me! Thanks!!
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32 views

Give a general formula in terms of $n$ for the determinant of the following matrix.

Let $M_n$ denote the $n$ x $n$ matrix over $\mathbb{R}$ of which the entry in the $i$-th row and the $j$-th column equals $1$ if $|i-j|\leq 1$ and $0$ otherwise. For example: $M_6=$ \begin{pmatrix} ...
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Vectors transformation without using a linear regression or a neural network

Is there any way to do a regression (transforming a set of high dimension vectors ($dim=400$, $number=500$) from a state $A$ to a state $B$) without using a linear regression or a neural network. I ...
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Prove convexity of function over space of positive definite matrices

I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. What I have done: It seems like this problem could be tackled by considering the ...
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Algebraic expression of a regression matrix

Let's say I'm doing a multivariate regression between a set of input $n$-dimension vectors (noted by the matrix $X=\{X_1,X_2,...X_m\}$) and a transformed version (noted by the matrix ...
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1answer
28 views

For $n>1$ $\hat x$, $p$ & $P$ [on hold]

For $n = 1$, $$\hat x = \frac{a^Tb}{a^Ta}, \quad p = a\frac{a^Tb}{a^T a}, \quad P = \frac{aa^T}{a^Ta}$$ What are these formulas when $N>1$?
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$Ax=b$ what does solving it mean?

We have been going through how to solve the system of equations known as $Ax=b$. Where $A$ is a matrix, $x$ is a vector and $b$ is a vector. I understand that if we have $A$ and $b$ we must find out ...
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2answers
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be J be a matrix so: JJ^-1 = I and A a matrix so: A^t JA = J. prove that A is invertible so that AA^-1 = I

be J be a matrix so: $JJ^{-1}$ = I and A a matrix so: $A^tJA^ = J$. prove that A can invertible so that $AA^{-1} = I$ the big question here is: what are the properties of A transpose, that allows ...
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1answer
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A is normal if and only if every matrix unitarily equivalent to A is normal

I need to prove that if $A$ and $B$ are unitarily equivalent, then $A$ is normal if and only if $B$ is normal. The proof is as follows: Suppose $A$ is normal and $B = U^*AU$, where $U$ is unitary. ...
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1answer
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How to find the basis?

So if there is a n-dimensional space (x_1, x_2, ..., x_n) over a field, How can I find a basis which together with (c_1, c_2, ..., c_n) (where all c_i's are non-zero constants from this field) forms a ...
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1answer
34 views

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ ...
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Proving or disproving the existence of a vector space

Suppose the vector space $V$ is spanned by $(1,0,1,0), (0,1,0,1), (1,1,0,0)$. Is it possible to find a subspace U of $\mathbb R^4$ such that $V \subsetneq U \subsetneq \mathbb R^4$? Note that ...
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1answer
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Derivation of Simple Projectile Motion with Drag

Given the initial velocity $v_0$ and angle $\theta$ of a projectile on the ground, using Newton's second law and the acceleration due to gravity $\mathbf g=\left\langle0,-g\right\rangle$, I was able ...
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iid Gaussian random matrix $A\in M_n$ has full rank with probability 1?

I want to prove that: iid Gaussian random matrix $A\in M_n$(I mean whose elements are iid Gaussian) has full rank with probability 1 Below is my consideration: $$1-P(\text{full ...
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3answers
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Diagonalize Matrix A

I am told to diagonalize Matrix A i solved for P and P inverse ...
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1answer
44 views

How to take the derivative of a matrix with respect to itself?

Could someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix
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If the Unit Vectors are equal, Are the directions equal too?

Given that the unit vector of x = unit vector of y, can we conclude that the Direction (or Sign) of x is always equal to Direction of y?
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How do I construct a Schauder basis of $l^2$ with the set $\{v_n\}

How do I construct a Schauder basis of $l^2$ with the set $\{v_n\} \subset l^2$, where $v_n=\frac{1}{\sqrt{2}}(e_n-e_{n+1})$ if $n$ is odd and $v_n=\frac{1}{\sqrt{2}}(e_n+e_{n-1})$ if $n$ even. Note ...
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38 views

How was step 1 done in Gaussian Elimination?

Suppose I have matrix $B:= \begin{bmatrix}4 & -2 & 2\\-2 & 5 & 3\\ 2 & 3 & 7 \end{bmatrix} $ Performing Gaussian Elimination we get: EDIT corrected mistake. I mistakenly ...
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1answer
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Is the function continuous that maps a vector to the coefficients of its expansion in a basis?

Let $V$ be a finite-dimensional vector space with basis $e_1, e_2, \ldots, e_n$. Consider the function $f:V\rightarrow\mathbb{R}^n$ defined such that, for each $v\in V$, $f(v)=(a_1,a_2,\ldots,a_n)$ ...
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an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? ...