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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Subspace of a finite dimensional dual space that separates points

As i was reading i came across a statement of this kind: If $V$ is a finite dimensional vector space and E is a subspace of its dual $V^*$, $E\le V^*$ and $E$ separates points than $E=V^*$. I ...
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2k views

How to solve Ax=0.. with 4 unknowns and 4 linear equations

I am trying to solve 4 linear equations for a 3D triangulation problem to create a function in matlab code. I have 4 equations such as aX + bY + cZ + dW = 0 eX + fY + gZ + hW = 0 iX + jY + kZ + lW ...
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1answer
27 views

Matrix Composed of Traces of Linear Independent Set of Matrices

I got the following problem: Let $S=\{A_1,A_2,...,A_k\} \subseteq \mathbb{M^R}_{n\times n}$ be a linear independent set of $k$ real $n \times n$ matrices with respect to the standard matrix inner ...
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26 views

Show that a set S is a subspace.

I have this assingment: Show that $ S = \{A \in \mathbb{R}^{3\times 3} | A^T = -A\} $ is a subspace in $ \mathbb{R}^{3\times 3} $ How do I do that? In the answer is says just: $$ 0\in S, then \; ...
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33 views

Direct sum decomposition of $l_2$

Let $X=(V,E)$ be a finite graph and a linear operator $\nabla: l_2(V) \to l_2(E)$ given by the formula $(\nabla f)(x,y)=$ \begin{cases} f(x)-f(y) &d(x,y)=1\\ 0 &\text{otherwise} ...
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51 views

Find sufficient and necessary conditions on $a$ and $b$ such that the equation have solutions of the form $(x,y)=(0,y)$

Let us consider the following equation $$ax+by=0$$ where $x$ and $y$ are the unknown variables and $a$ and $b$ are constants. My question is: Find sufficient and necessary conditions on $a$ and $b$ ...
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129 views

If A unitary matrix and orthogonally diagonalizable why there is a basis in whichthe linear trans. matrix is diagonal?

If $A$ is a $n\times n$ unitary matrix (above the complex field) and is orthogonally diagonalizable, why does it mean that the is an orthonormal basis $\mathbb C$ in which the matrix that represent ...
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34 views

how to construct $A$ [duplicate]

Construct a $2$ $\times$ $2$ matrix $A$ ($\neq$ $I$) with entries from $\mathbb{R}$ such that $A$$^3$=$I$. First give me some hint. How to construct this kinds of matrices, is there any rule...
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1answer
4k views

Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...
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42 views

Does this bidiagonal matrix have a name?

Do bidiagonal matrices like the one below have a name? $$\left(\begin{array}{cccc} a_1 & a_2 & & & \\ & a_2 & a_3 & & \\ & & a_3 & a_4 & \\ ...
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43 views

Show that, if $M$ is an $nxm$ matrix that $|M|^2$ = $tr(M^TM)$

No Idea how to do this one. All I know is that, if a matrix is diagonal, then the determinant equals the trace.
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216 views

Linear algebra : eigenvalues of an integral operator on polynomials

Consider the linear transformation $$ T : \left\{ \begin{array}{ccc} \mathbb{R}_n[X] & \to & \mathbb{R}_n[X] \\ P & \mapsto & \int_0^1 (X + t)^n\,P(t)\,dt \end{array}\right. $$ where ...
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504 views

Units of eigenvalues [closed]

Suppose you have the system $\bf x' = \bf Ax$, where $\bf x$ is a vector and $\bf A$ is a matrix. What are the units of the eigenvalues of $\bf A$? I think they should be $1/t$ but I'm not sure how to ...
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46 views

Inverse of sum of fractions

I'm interested in the inverse of a finite sum of fractions. eg: $$ \large{\frac{1}{\sum_{i=1}^{n} \frac{a_i}{b_i} }}$$ For $a_i, \ b_i \in \mathbf{R}$. Specifically, can this be expressed in terms ...
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1answer
1k views

Determine if b is in the span of the other given vectors.

Determine if the vector $b$ is the span of the other given vector. If so, write $b$ as a linear combination of $a$ a=$\begin{bmatrix} 3\\ 5\end{bmatrix}$ b=$\begin{bmatrix} 9\\ -15\end{bmatrix}$ I ...
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967 views

Adding Elements to Diagonal of Symmetric Matrix to Ensure Positive Definiteness.

I have a symmetric matrix $A$, which has zeroes all along the diagonal i.e. $A_{ii}=0$. I cannot change the off diagonal elements of this matrix, I can only change the diagonal elements. I need this ...
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1answer
68 views

Matrix related question

My question is why new equation is named as 3rd equation after adding equation first and third. Why don't we name it as first equation and then write in the place of first equation in the new ...
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3answers
181 views

eigenvalue of block matrix in terms of original matrix

A is a $4*4$ matrix with eigenvalues $\lambda_A$. Consider a block matrix $B = \left( \begin{array}{ccc} A & I \\ I & A \end{array} \right) $. Then how can we find eigenvalue $\lambda_B$ of ...
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1answer
789 views

Find the equation of the plane that passes through a point and parallel to another plane

Find the plane given that it passes through the point $P = (0, -2, 5)$ and parallel to the plane $6x - y + 2z = 3$. If we're given $P$ in the equation is $$n \cdot r = n \cdot p$$ where $r = ...
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55 views

How to calculate the hight by number of nodes

Imagine that I have something like following structure and I keep adding more to it, so the level 1 has only one node and level 2 has 2 and level n had n node, how can i calculate the n from the total ...
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1answer
145 views

orthogonal base in inner product

I tried to solve this problem: Let $V$ be an inner product space over a field $F$. And let $u_1, \ldots, u_k$ be linearly independent vectors such that: $\forall \space v\in V: ...
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3answers
440 views

Dimension of sum and intersection of vector space.

I am trying to understand the proof of the following: Suppose $U,W$ are vector subspace of $V$, then $\dim (U+W)+\dim (U \cap W)= \dim (U) +\dim (W).$ The proof goes like this: Let $S: V \rightarrow ...
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138 views

Orthogonal polynomials on $[0,1]$

Are the orthogonal polynomials for the standard $L^2$ product on $[0,1]$ well-known? I couldn't find anything upon a quick web search.
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1answer
29 views

Let $A \in$ Mat$_{n,n}(\mathbb R)$ be invertible and have the property that every row sum is $1$. Prove $A^{-1}$ has the same row sum property.

Let $A \in$ Mat$_{n,n}(\mathbb R)$ be invertible and have the property that every row sum is $1$. Prove $A^{-1}$ has the same row sum property. I have tried the following: $\sum_{j=1}^n ...
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2answers
474 views

Why $\mathbf{0}$ vector has dimension zero?

According to C.H. Edwards' Advanced Calculus of Several Variables: The dimension of the subspace $V$ is defined to be the minimal number of vectors required to generate $V$ (pp. 4). Then why does the ...
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30 views

$X \in Mat_{n,n}(\mathbb F)$. Prove $XA = AX \ \forall A \in Mat_{n,n}(\mathbb F) \iff X = sI, s \in \mathbb F$

$X \in Mat_{n,n}(\mathbb F)$. Prove $XA = AX \ \forall A \in Mat_{n,n}(\mathbb F) \iff X = sI, s \in \mathbb F$ Proving that $X = sI \Rightarrow XA = AX$ is easy by manipulation the multiplication ...
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1answer
100 views

System of linear equations over Finite Field with restriction on variables

Given two vectors $x$ and $y$, where each element $x_i, y_i$ is from a finite field. I have the restriction that for each of these variables only about half of the elements of this finite field are ...
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189 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
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43 views

Prove if $A \in Mat_{n,n}(\mathbb F)$ is both symmetric and skew-symmetric then $A=0$

Prove if $A \in Mat_{n,n}(\mathbb F)$ is both symmetric and skew-symmetric then $A=0$ I know $A^T = A = -A \Rightarrow A = -A \Rightarrow A_{i,j} = -A_{i,j}$. Since $\mathbb F$ is a field we have ...
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1answer
46 views

What's self-consistent matrix?

I'm reading a paper and it mentioned the following: "one obtains this self-consistent matrix formulation:" and then they mentioned a formulation. But I couldn't understand the meaning of a ...
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1answer
71 views

Jordan forms for linear operator

I was reviewing linear algebra and came across the following problem in Artin: The characteristic polynomial for a linear operator $T$ is $(t-\lambda)^5$. And the rank of $(T-\lambda I)$ is 2. What ...
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1answer
38 views

Defining a basis for a transformation matrix in the complex numbers

I have to show that $f: \mathbb{C} \rightarrow \mathbb{C}$ by $f(z) = \operatorname{Re}(z)$ is a linear transformation (easy) and give a matrix of transformation (easy) I say its easy because I've ...
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0answers
100 views

solving underdetermined system of equations

I have an equation of the form $$U = T_1 \times S_1 + T_2 \times S_2 + T_3 \times S_3$$ where $U$ = utilization $T_i$ = throughput of $i$ $S_i$ = service time of $i$ constraint, ...
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34 views

In the Schwarz inequality, what happens when $a$ and $b$ lie on the opposite sides of the origin?

The Schwarz inequality is an equality if and only if $b$ is a multiple of $a$. The angle is $\theta = 0$ or $\theta = 180$, so $\cos \theta = 1$ or $-1$. In this case $b$ is identical with its ...
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Matrix $A=B+C$ with $B$ symmetric and $C$ antisymmetric

I am stumped on a question and am looking for some guidance on how to get it done. The problem gives you: $x_1 = \begin{bmatrix}9&-4&-2 \\-9&6&-3 \\10&-3&9\end{bmatrix}$ ...
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1answer
29 views

Show that $ \left \| p \right \|=\left \| b \right \|\cos \theta$ from $p=\hat{x}a = \frac{a^{T}b}{a^{T}a}$

Since $\left||a\right\|^{2} = a^{T}a$ and $a^{T}b = \cos \theta \left ||a\right|\left||b\right|$, then $p=\hat{x}a = \frac{\cos \theta \left ||a\right|\left||b\right|}{\left||a\right\|^{2}}a$. I ...
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1answer
34 views

Linear Algebra Orthogonal Distance Question

Here is the question: Let $V$ be a vector space with an orthogonal basis $\{\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3,\mathbf{u}_4\}$, $W=\mathrm{span}\{\mathbf{u}_1,\mathbf{u}_3\}$ and ...
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2answers
59 views

Matrix of transformation for reflections

I want to find two matrices of transformations from $\mathbb{R}^3$ to $\mathbb{R}^3$: reflection over $x=y$ reflection over $y=z$ how do I do this?
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58 views

Directional derivatives, linear maps, and uniform convergence

The Exercise Let $f(x,y)=x$ if $|y|>x^2$ and $f(x,y)=0$ otherwise. Show that all the directional derivatives of $f$ exist at the origin but there does not exist a linear map $D$ such that ...
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1answer
305 views

Proving that $a$ is eigenvalue of $p(T)$ iff $a=p(\lambda)$ for eigenvalue $\lambda$ of $T$

The solution is below, I just do not understand why if: $p(T)-aI$ is not injective, then $T-\lambda_jI$ is not injective for some j either. Also, what does repeatedly applying T to both sides ...
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1answer
48 views

Prove the existence of the following

Let $V$ be a vector space over the field $F$. Prove that for every subspace of $V$, $W$, there exists a subspace $U$, such that $W+U=V$ and $W \cap U= \left\{ \mathbb{O}_V \right\}$ Where ...
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2answers
71 views

Show that ${\boldsymbol v}_1$ and ${\boldsymbol v}_2$ are linearly independent [duplicate]

Show that if $r_1 \neq r_2$, the vectors (functions) $${\boldsymbol v}_1 = \exp(r_1t),\,\,\,\,\,\,{\boldsymbol v}_2 = \exp(r_2t)$$ are linearly independent in the space of continuous functions ...
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5answers
546 views

Is it acceptable to solve hypothetical statements in Linear Algebra using actual numbers?

I'm taking a Linear Algebra course this semester where we must prove/disprove hypothetical statements. So I'm wondering, is it alright to show that certain theorems hold or not using examples with ...
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1answer
69 views

Basis with small $\ell_{\infty}$ norm for Subspace

At a high level, I'm interested in finding an orthonormal basis for a $d$-dimensional subspace $U \subset \mathbb{R}^n$ with small entry-wise $\ell_{\infty}$ norm. In particular, suppose $v_1, ...
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2answers
2k views

linear independence and reduced row echelon form

If I can write vectors $(2,0,0)$ ,$ (1,-1,0)$ and $(0,1,1) $ as $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}$ using reduced row echelon form does this means that they ...
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2answers
155 views

Does first isomorphism theorem hold in the category of normed linear spaces?

Consider the category of normed linear spaces over $\mathbb{C}$ with bounded linear maps as morphisms. If $M\subset X$ is a subspace, then the quotient space $X/M$ has a map $\|x+M\|: = \inf_{y\in ...
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2answers
267 views

Positive matrix and positive vector

Let $A \in \mathbb{R}^{n \times n}$ be a non-negative matrix, i.e. $A_{i,j} \geq 0$ $\forall (i,j)$. Let $x \in \mathbb{R}^n \setminus \{0\}$ be a non-negative vector, i.e. $x_i \geq 0$ $\forall i$. ...
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1answer
115 views

Is $(V_1\otimes\cdots\otimes V_k)^\ast \simeq V_1^\ast\otimes \cdots \otimes V_k^\ast$ true for infinite dimensional spaces?

Suppose $V_1,\dots,V_k$ are vector spaces of finite dimension. Then I could prove easily that $(V_1\otimes\cdots\otimes V_k)^\ast\simeq V_1^\ast\otimes\cdots\otimes V_k^\ast$. My proof was like that: ...
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2answers
97 views

Linear Algebra - Given the Jordan form of $A \in Mat_7(\mathbb F)$, find Jordan form of $A^2+A+I_7$

Given that the jordan form of the matrix $A \in Mat_7(\mathbb F)$ is: $\begin{pmatrix} J_2(1) &\cdots &0\\0& \cdots J_3(1) \cdots &0\\0&0& \cdots J_2(2)\end{pmatrix}$ Find ...
1
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1answer
46 views

Algorithm to generate normal matrices at random

I would like to generate normal matrices by an, say python, algorithm, that produces normal matrices distributed evenly in the limit of large n. I would not like to be restricted to Hermitian matrices ...