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 ...

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Find parametric vector form of a cartesian equation under a specific condition

Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1 ...
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5 views

$|f(x)-f(y)|\leq u(a,b)|x-y|^t$ for all $a\leq x,y \leq b$. For what values of $t$ is $V_t$ a subspace of $\mathbb{R}^\mathbb{R}$?

For any real number $0<t\leq 1$, let $V_t$ be the set of all functions $f\in \mathbb{R}^\mathbb{R}$ satisfying the condition that if $a<b$ in $\mathbb{R}$ then there exists a real number ...
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16 views

If $A: M \to M$ then $M$ is $A$-invariant subspace and $A $ is an endomorphism?

Just straightening out the terminologies here... Given If $A: M \to M$ then $M$, $M$ some subspace of a vector space, is the following statement equivalent: $M$ is a $A$-invariant subspace $A $is ...
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22 views

Do addition and multiplication define a structure of a field? [duplicate]

I am taking an advanced linear algebra course for my Masters but never took linear in undergrad so please realize I know little to nothing about these topics. Question: Let r exist in R and 0 not ...
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26 views

Schwarz equation proof help!!

Hello, In this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for ...
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1answer
21 views

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$.

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$. The multiplicative inverse is $(1,0)$. I need to show that ...
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1answer
13 views

transformation matrix between two different basis

I am working on this problem:- A rectangular coordinates $(x,y,z)$ are given in terms of new coordinates $(q_1,q_2,q_3)$ by :- $x=q_1 +q_2 \cos(\theta)$ , $y=q_2 \sin(\theta)$ and $z=q_3$. where ...
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1answer
44 views

Proof of a Vector Space

Let $F$ be a field and let $(V, +, F)$ be a vector space over $F$. If $W_1$ and $W_2$ are subspaces of $F$, prove that $W_1 - W_2 = \{v \in V | v = w_1 - w_2 \text{ for some } w_1 \in W_1, w_2 \in W_2 ...
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1answer
17 views

If $A,B \in \mathcal{M}_n$ and $AB=BA=\mathbb{O}_n$ prove that $(A+B)^ν = Α^ν + Β^ν$.

I have one exercise in Linear Algebra and I would like to know if my solution is correct. If $A,B \in \mathcal{M}_n$ and $AB=BA=\mathbb{O}_n$ prove that $(A+B)^ν = Α^ν + Β^ν$. My first thought is ...
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23 views

Understanding first part of dual basis proof

The textbook I'm reading attempts to proof the following: given $\left\{v_1, \ldots, v_n \right\}$ a basis for a vectorspace $V$ over $K$, there exists a basis $\left\{ \phi_1, \ldots, \phi_n ...
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7 views

Legendre transform is everywhere finite iff f grows faster than 2-norm

Let $f:\mathbb{R}^n \to \mathbb{R}\cup \{ \infty \}$ be convex. Its Legendre transform is $f^* (d):=\sup_{x\in \mathbb{R}^n}(d^Tx-f(x))$ Show $f^*(d)<\infty$ $\forall d\in \mathbb{R}^n$ iff ...
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2answers
19 views

How to solve for the matrix $X$ in the following equation $AXB + X = CD$

How to solve for the matrix $X$ in the following equation $AXB + X = CD$? $A$ and $B$ are full rank symmetric matrices, and there is no structure to $CD$. $CD$ just could be $C$.
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1answer
28 views

A question on numerical range

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrum of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
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3answers
80 views

Trouble with understanding dual space $V^{*}$.

I've just began the study of linear functionals and the dual base. And this book I'm reading says the dual space $V^{*}$ may be identified with the space of row vectors. This notion seems very ...
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35 views

Proof of Simple Properties of Volume

Let $e_{1},\ldots,e_{n}$ be vectors in $\mathbb{R}^{n}$. Define a parallelepiped $P$ to be a translate of the set $$\left\{x\in\mathbb{R}^{n} : x=t^{1}e_{1}+\cdots+t^{n}e_{n}, 0\leq t^{i}\leq ...
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2answers
32 views

Let $V=\bigcup_{i=1}^n W_i$ where $W_i$ s are subspaces of a vector space $V$. Show that $V=W_r$ for some $1 \leq r \leq n$.

Let $V=\bigcup_{i=1}^n W_i$ where $W_i$ s are subspaces of a vector space $V$ over an infinite field $F$. Show that $V=W_r$ for some $1 \leq r \leq n$. I know the result "Let $W_1 \cup W_2$ is a ...
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1answer
27 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
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27 views

How numerical radius help us to conclude an operator is normal and partial isometry?

In Furuta's book, "Invitation to Linear Operators" there is a theorem, theorem 2 in 3.7.3, that says: If $T^k=T$ for some integer $k\ge 2$ and if $w(T)\le 1$, then $T$ is the direct sum of a unitary ...
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1answer
18 views

Linear transformations for fixing the line $y = 0$

The professor says that the subgroup for "stabilizing" the line $y = 0$ is $$A = \begin{bmatrix} a & c \\ 0 & d \end{bmatrix}$$ because in order to fix the first basis vector, $b = ...
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14 views

If $L(v_1)=L(v_2)=L(v_3)=w_1$ then what is the $rank(L)$?

it is an elementary question. $L:V\to W$ is a linear transformation and $S=\{v_1,v_2,v_3\}$ is an ordered base of V, and $T=\{w_1,w_2\}$ an ordered base of W. If $L(v_1)=L(v_2)=L(v_3)=w_1$ what is ...
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3answers
44 views

Orthonomal bases and cross products

I want to show that if I have an orthonormal basis of $\mathbb{R}^3$, say $\{\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}\}$, and if $\boldsymbol{u} × \boldsymbol{v} = \boldsymbol{w}$, then we have ...
5
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1answer
30 views

Classification of all conjugacy classes of $GL_2(\mathbb{R})$, $GL_2(\mathbb{Q})$.

Give a classification of all conjugacy classes in the following groups. $GL_2(\mathbb{R})$ $GL_2(\mathbb{Q})$ My progress so far. If the characteristic polynomial splits, the matrix ...
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14 views

Matrices of Ordered Bases

Let $V$ be a real finite-dimensional vector space and $T : V → V$ be a linear map. Let $E$ be a basis of V . What does it mean to say that $A$ is the matrix of $T$ with respect to $E$. Let $S : V → V$ ...
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17 views

How are closed form solutions for eigenvalues constructed from sines and cosines?

For example a size N 3-point finite difference scheme has eigenvalues $\lambda_j = 2+cos(j\pi/(N+1))$. How is this determined? I know the Gershgorin circle theorem, but this is not what I am looking ...
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2answers
44 views

generalized Cauchy-Schwarz inequality

How to prove $A'B(B'B)^{-1}B'A \leq A'A$, where $A$,$B$ are $n\times k$ matrices and $B'B$ is assumed to be positive definite? I don't see why it is a Cauchy-Schwarz inequality.
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1answer
17 views

Approximate solution to a matrix equation

Let $A$ and $B$ be $n \times m$ matrices. I am looking for a $m \times m$ matrix $X$ which would be an approximate solution to the equation $AX = B$ (an exact solution is very unlikely to exist). More ...
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2answers
31 views

$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
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2answers
48 views

Can such an “orthogonal” matrix exist?

I know that the definition of an orthogonal matrix is that $A \in \mathbb R^{n \times n}$ is orthogonal if $AA^T = A^T A=I$, no problem with that whatsoever. My question is this - Why only square ...
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12 views

What are the (more general) conditions for consistency in a system of linear equations?

Usually, when speaking about conditions for the existence of solutions to linear equations of the form $A x = b$ (with $A \in \mathbb{R}^{n \times n}$ and $x , b \in \mathbb{R}^n$), one is interested ...
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2answers
30 views

Is $V = \{(x,y,z)\in \mathbb{R}^3:\ x+y >1 \}$ a subspace?

Prove whether the following subsets of $\mathbb{R}^3$ are subspaces : (a) $$V = \{(x,y,z)\ \in \mathbb{R}^3:\ x+y >1 \ \},$$ I think that this is not a subspace as the zero vector does not ...
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2answers
18 views

tangent plane for y^x at point (2,1)

I test my answer using wolfram alpha pro but it gets a different result to what I am getting. This is homework. My result is z= 2(y-1) partial derivative with respect to y is ...
2
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1answer
18 views

How to find the minimum sum of unknown variables that is a solution to a system of two linear equations?

I'm trying find the minimum sum of $x_{1} + x_{2} + ... + x_{n}$ where these are a solution to a linear system of two equations. System of linear equations in general form: $$ a_{11}x_{1} + ...
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19 views

algebra questions [on hold]

On the weekend you played Rugby League. You scored three tries, made two conversions and one field goal. How many points did you gain for your team in total? Write a general equation for this problem. ...
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2answers
57 views

Number of matrices $A \in M_n(\mathbb{F}_q)$ where $A^2 = 0$.

What is the number of matrices $A \in M_n(\mathbb{F}_q)$ for which $A^2 = 0$ (as a function of $n$ and $q$)?
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5answers
35 views

Seemingly impossible problem involving linear combination of vector components.

Express $\langle 4, -8 \rangle$ as a linear combination of $\vec{u}$ and $\vec{v}$, given $\vec{u}=\langle 1,1 \rangle$ and $\vec{v}=\langle -1,1 \rangle$. So, I set up: $\vec{i}=\langle 1,0 \rangle$ ...
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2answers
37 views

Is $V$ a simple $\text{End}_kV$-module?

Let $V$ be a finite-dimensional vector space over $k$ and $A = \text{End}_k V$. Is $V$ a simple $A$-module?
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1answer
47 views

How to solve these equations?

How to solve these equations for a, b, c and x? I have the following: $ 2a+b+c = 1$ $a = (a+b)x + 0.25(a+c) $ $a=(a+c)(1-x)$ $b=a(1-x)+c(x-0.25)$ $c=b(1-x)+a(x-0.25)$ I tried, but ended ...
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17 views

Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2 $ and we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
5
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0answers
37 views

Square root of differentiation

Let $T=d/dx$ be the differentiation operator on vector space $V=C^{\infty}(\mathbb{R})$, the space of real (complex) valued smooth maps on real line. To what extent, all subvector space ...
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2answers
15 views

What am I doing wrong in finding the orthogonal projection of a vector onto the subspace V?

Let $V∈ℝ^5$ be the subspace $V=span{(2,0,0,0,1),(0,2,0,3,0)}$ and let $w=(0,0,-4,-1,-1)$. Find the orthogonal projection of $w$ onto V,using exact values in your answer. My Approach Let the ...
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1answer
23 views

Finding Marginal Density functions with $Y\sim N_4(\mu,\Sigma)$

Suppose $Y$ is $N_4(\mu, \Sigma)$ where $$\mu = ( 1,2,3,-2)'$$ and $$\Sigma =\begin{bmatrix} 4& 2& -1& 2 \\ 2& 6& 3& -2 \\ -1& 3& 5& -4 \\ 2& ...
3
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1answer
27 views

Prove $v,w\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and dependent when $p=3$

I need to prove that $\{v=(6,9),w=(7,8)\}\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and linearly dependent when $p=3$. The problem is my freshman algebra course did not cover rings and ...
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1answer
15 views

About finding the common diagonalizing similarity transformation.

Say I have $2k$ matrices $M_{a_1b_1}$, $M_{a_2b_2}$,..,$M_{a_kb_k}$ and their negatives. Here $M_{a_ib_i}$ is such that it has $0$ everywhere except that it has $1$ at $(a_i,b_i)$ and $(b_i,a_i)$ ...
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2answers
37 views

Prove function space is linearly independent.

Let $V$ the space of all funcions $f:Ŗ\rightarrow R$. Prove that the ten functions defined by $x\rightarrow |x-1|$,$x\rightarrow |x-2|$,....,$x\rightarrow |x-10|$ are linearly independent. I need ...
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1answer
14 views

Given $k$ distinct linear operators, prove such an $\alpha$ exists

I have $k$ distinct linear operators $\{\phi_i\}$ which act on $V$, a vector space on some number field $K$ (in the sense that $\Bbb Q$ is the smallest possible one). Now I have to prove that there ...
1
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1answer
10 views

Isolate Costs in NPV equation

Hey can anyone help with this? This is the classic NPV equation: NPV = -CapEx + ∑ (Revenue − Costs) / (1+Discount)^i The partial sum is from i = 0 to n years. For my purposes all the elements ...
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2answers
60 views

Prove that three points define a unique parabola

How do we prove that there is always a unique parabola (with equation $y=ax^2+bx+c$) that passes through 3 distinct points $P_1 (p_1,q_1), P_2 (p_2,q_2), P_3(p_3,q_3)$ ? If I choose to use matrices ...
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0answers
24 views

Unitary matrix question [on hold]

Let $A=\begin{pmatrix}1&4\\ 2&5\\ 3&6\end{pmatrix}$. Find $B$ such that $B^*B=I$ (identity matrix) and $Im A= Im(B)$ I know that the $Im(A)$ is the set of all possible linear ...
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0answers
25 views

Checking if this set is a vector sp.

Question: Define a set $V=\{(x,y):x,y\in\Bbb R\}$. For any two elements $u=(u_1,u_2),v=(v_1,v_2)$ in $V$ and $t\in\Bbb R$, addition and scalar multiplication as, ...
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0answers
15 views

I don't understand this definition of vector positivity in my linear algebra text

I don't understand why they say that the magnitude of v is greater than or equal to zero and then go on to say the magnitude of v is equal to zero if the vector is equal to zero. Shouldn't they use ...