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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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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7 views

Complex projections order in inner product

So the complex projection is defined as $$proj_\vec{u}^{} \vec{v}^{} = \frac{<\vec{v}^{},\vec{u}^{}>}{<\vec{u}^{},\vec{u}^{}>} \vec{u}^{}$$ with complex inner product. I was wondering if ...
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16 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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2answers
24 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
22 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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16 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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1answer
23 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
42 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
25 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
35 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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11 views

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
14 views

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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13 views

A reduction of Cayley hamilton to the complex case

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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22 views

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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0answers
21 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
120 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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33 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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2answers
32 views

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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1answer
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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10 views

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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13 views

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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0answers
14 views

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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4answers
96 views

$Ax=b$ 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
29 views

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
15 views

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
45 views

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
32 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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1answer
23 views

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
28 views

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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2answers
12 views

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
18 views

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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20 views

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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1answer
17 views

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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1answer
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
17 views

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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1answer
23 views

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 ? ...
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25 views

Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ...
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19 views

If $\vec{v}$ is any non-zero vector perpendicular to $\vec{u}$, show that $\vec{v}$ is an eigenvector of $S$ [on hold]

I have the following problem.. I solved the first one however i can't find out how to solve the second (b). Suppose $\vec{u}$ is a unit row-vector in $\mathbb R^n$ , and $A=uu'$ matrix. (a) ...
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60 views

$x-y-2z=0$ find a perpendicular vector

Why is the vector $e=(1,-1,-2)$ ?
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1answer
27 views

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
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1answer
33 views

Bound spectral radius of a certain matrix.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi = \text{diag}(\xi)$ to be a diagonal matrix with a principal left eigenvector $\xi$ of $P$ (i.e. ...
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1answer
31 views

The dimension and basis of the set $F = \{(a+3b,a-b,2a-b,4b)| a,b \in \mathbb{R}\}$

Let $F = \{(a+3b,a-b,2a-b,4b)| a,b \in \mathbb{R}\}$ Show that F is a subspace of $\mathbb{R}^4$; Find a basis for F; Find the dimension of F. I have part A completed and showed ...