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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A question in numerical range.

In this question, is this true that (if $0 \notin W({A_m})$ then $W(P(\lambda ))$ is bounded)?
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Evaluation of $5\times 5$ determinant

The following $5\times 5$ det. comes from a Russian book. I don't want to expand the det. rather than do some operations on it and extract the result. Prove: $$\begin{vmatrix} -1 &1 &1 ...
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Solving simultaneous equation when one changes the other

You have two points given by $(w,x)$ and $(y,z)$. Can $(w,x)$ be transformed into $(y,z)$ by only performing the operations $(w+x,x)$ and $(w,x+w)$? $w,x,y,z$ are integers greater than or equal ...
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2answers
17 views

simplify expression with indices

Given the following expression: ${{{x^{2\over3}} + {x^{3\over4}} + 2}\over x^4}$ I can split this out to: ${{{{x^{2\over3}}\over x^4}} + {{x^{3\over4}}\over x^4} + {2\over x^4}}$ I can then divide ...
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2answers
22 views

Proving that an operator $T$ on a Hilbert space is compact

Let $H$ be a Hilbert space, $T:H \to H$ be a bounded linear operator and $T^{*}$ be the Hilbert Adjoint operator of $T$. Show that $T$ is compact if and only if $T^{*}T$ is compact. My attempt: ...
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2answers
42 views

Why is it called linearly independent?

For a system of linear equations in $\Bbb R^n$ to be linearly independent, there must be a unique solution to the system (at least I'm pretty sure that's true). There are definitely other definitions, ...
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0answers
26 views

Proof with Linear Transformation

Let $u$ = ($u$1, $u$2, $u$3) ∈ $\Bbb R^3$ and $v$ = ($v$1, $v$2) ∈ $\Bbb R^2$ be non-zero (row)vectors. Define F : $\Bbb R^3$ → $\Bbb R^2$ by F($x$) = ($u$ · $x$)$v$. (a) Show that ker F = ...
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19 views

A question in numerical range

Suppose ${A_j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a complex ...
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1answer
20 views

Existence of a surjective compact linear operator on an infinite dimensional Banach space

Does there exist a surjective compact operator $T:l^{\infty} \to l^{\infty}$ ? Even though this might be tagged as a repeat question, i still have some doubts that i would like to clarify. My ...
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2answers
22 views

Proving that an operator is Compact

I have to check that the following operator $T$ is compact: Define $T:l^{2} \to {l^2}$ by $Tx=y=(\eta_{j})$, where $x=(\zeta_{j})$ and $(\eta_{j})=\sum_{k=1}^\infty \alpha_{jk} \zeta_{j}$ and ...
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Find a linear transformation by $\operatorname {Im}(T)=\ker(T)=\operatorname{Span}\{1-x,x-x^3\}$

Any hints how to find exact form of $$ T:\Bbb R_4[x] \to \Bbb R_4[x] $$ $$ T(ax^3+bx^2+cx=d) $$ $$ a,b,c,d\in\Bbb R $$ given $$ \operatorname {Im}(T)=\ker(T)=\operatorname{Span}\{1-x,x-x^3\} $$
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3answers
35 views

What does Linear mean in Linear Space (Vector Space)

The course I'm taking defines a vector space or Linear space as The vector space $\mathbb{R}^n$ has a linear structure with two features: vector addition and scalar multiplication" What does it ...
3
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1answer
36 views

Finite series for the matrix exponential

It is well know that any analytic function of an $n \times n$ real/complex matrix $f(A)$ can be expressed as linear combination of the first $n$ powers of $A$ by the Cayley-Hamilton theorem. Is it ...
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1answer
56 views

Prove that $\sum_{i,j} \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0$

Let $v_1 \dots v_n, w_1 \dots w_n \in H$ an inner product space. I am trying (unsuccesfully) to show that $$ \sum_{i,j=1}^n \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0 .$$ Any hints?
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Relation between Image and kernel of matrix of sufficient statistics

Suppose we have a parametric statistical model formed by $p_x = \frac{L(\zeta, x)}{\sum_{y\in \mathcal{X}}L(\zeta,x)}$ where $\mathcal{X}$ is the sample space, $L(\zeta,x) = \zeta^{T(x)} = ...
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What are some unanswered problems on vector algebra? [on hold]

I'm sorry for any mistakes.Math language is different from my country's so they may be so wrong tags. Thanks in advanced ^^
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11 views

$S(0,\varepsilon ) \Rightarrow F + S(0,\varepsilon ) = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$

Let $F \subseteq {\rm{C}}$ and $S = \left\{ {x \in C:\left\| x \right\| \le \varepsilon } \right\}$. Why does $F + S = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$? (where ...
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2answers
40 views

Prove the matrix $M$ is orthogonal if and only if $M^T= M^{-1}$

Prove the matrix $M$ is orthogonal if and only if $M^T= M^{-1}$ I know that I have to show $$M \text{ is orthogonal } \implies M^T = M^{-1}$$ and $$M^T = M^{-1} \implies M \text{ is orthogonal }$$ ...
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32 views

Understanding part of the proof of Spectral Theorem for symmetric matrices

I'm reading a textbook where the Spectral Theorem for symmetric matrices is proven. I understand almost everything about the proof except for one thing. The theorem is stated as follows: Theorem: Let ...
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1answer
26 views

steps involved in matrix algebra problem

If $E$ is a column vector, $\Sigma$ is an $n \times n$ symmetric matrix, Let : $$ A = E^T \Sigma^{-1}E \quad~~~~~~ B = E^T \Sigma^{-1}1 ~~~~~\quad C= 1^T \Sigma^{-1}1 \quad $$ Then let: $$ w = ...
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1answer
15 views

Compactness of a linear operator

The question is as follows: Show that a linear operator $T:X\to Y$ where $X$ and $Y$ are normed spaces is compact if and only if the image $T(M)$ of the unit ball $M\subset X$ is relatively compact ...
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0answers
32 views

Matrix multiplication of columns times rows instead of rows times columns

In ordinary matrix multiplication $AB$ where we multiply each column $b_{i}$ by $A$, each resulting column of $AB$ can be viewed as a linear combination of $A$. If however if we decided to multiply ...
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0answers
43 views

Does this binary operation have a name?

Does the following binary operator on abstract linear maps $A,B:\mathbb{C}^N \rightarrow \mathbb{C}^N$, have a name: $[\{A,B\}]:= AB^{\dagger} - BA^{\dagger}$ clearly, it is real bi-linear, but not ...
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18 views

vector 3d rotation of a cube

I have a cube which is rotated by plane you can see it in an example here. What am I trying to achieve is algorithm that tells what is the top, face and side after a rotation is performed. And also ...
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1answer
25 views

Verification for a block-determinant evaluation, and some further thoughts

First, I want some verification for the validity of my approach for this det evaluation question: If $A,B\in M_n(K)$, $K$ is a number field (in the sense that $\Bbb Q$ is the smallest possible ...
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1answer
32 views

A program to visualize Linear Algebra?

I am asking here because I believe you have some idea of a good visualizer 3d program to see what are really: eigenvectors, subspaces, rowspaces, columnspaces and just answers on normal matrix ...
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0answers
15 views

Linear dependence of rank one approximations.

In my research, I am faced with the following problem. Let $\mathbf{X}$ and $\mathbf{Y}$ be two nonzero $m \times n$ matrices with entries in some field $\mathbb{K}$. Let $\phi:\mathbb{K}^{m\times ...
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10 views

Some “facts” on oriented angles in the Euclidean affine space of dimension 2

Let $\mathcal{E}^2$ be an Euclidean affine space of dimension $2$ oriented by $R=(O,B=\{u_1,u_2\}$. Let $$\mathcal{r}=\{P \in \mathcal{E}_2 \text{ such that } \overrightarrow{AP} \in \langle u ...
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1answer
17 views

Matrix operation: putting the rows next to each other

I have a matrix $A$ of dimension $N\times K$, and want to find a way to convert it to a matrix $B$ of dimension $1\times NK$. For example: ...
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1answer
20 views

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let's take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P ...
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2answers
384 views

Any neat way to calculate this Vandermonde-like determinant?

Let $x_i,i\in\{1,\cdots,n\}$ be real numbers, and $s_k=x_1^k+\cdots+x_n^k$, I'm asked to calculate $$ |S|:= \begin{vmatrix} s_0 & s_1 & s_2 & \cdots & s_{n-1}\\ s_1 ...
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43 views

Matrix of a differential equation

I had recentely encounter my first exercise about merging matrices notions and differential equations functions, but after solving the differential equation, I don't know how to represent it in the ...
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1answer
21 views

Computationally inexpensive method to find a rotation which minimizes the norm of two tensor difference.

So I have two matrices ${\bf T}_1$ and ${\bf T}_2$, they are tensors in the sense that they can be built as $${\bf T} = \sum_{\forall i} a_i({\bf v_i}{\bf v_i}^T)$$ with positive real weights $a_i$ ...
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0answers
34 views

Linear Algebra Subspace question, set difference [on hold]

If $W_1$ and $W_2$ are subspaces of a vector space $V$, is $W_1 \setminus W_2$ ever a subspace of $V$? Why or why not? (Here $W_1 \setminus W_2$ denotes the set difference of $W_1$ and $W_2$: $W_1 ...
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0answers
19 views

How can we multiply partitioned matrices?

Im trying to perform the following product: Question: can we just think of each block as an element and perform the multiplication like matrix multiplication?
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0answers
51 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
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3answers
64 views

Inner Product, Orthogonality, and Coordinate Systems

I remember my professor saying there are certain advantages to using an orthogonal basis. One is that it's easy to determine the coordinates of a given vector. For example, we are familiar with the ...
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2answers
38 views

How did they derive the image from kernel?

I understand its something to do with the rank nullity theorem, but im not sure how they applied it to get the basis of the image. By my understanding, they took the leading entries of the rows of ...
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1answer
23 views

Linear equation and linear differential equations

I remember noting from an algebra class that $x$ and $y$ of a linear equation neither divide or multiply with each other which is somewhat clear from the forms of linear equations: General form of ...
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1answer
46 views

What value to choose as representative of 100 votes?

I'm trying to use many votes to cast one vote. There are only 4 options, and I'm trying to use "consensus" to decide which to pick. 100 people vote. They can vote 1, 2, 3, or 4 stars. Distribution: 1 ...
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1answer
35 views

Making sense of $ f(y) - f(x) = \int_{\tau = 0}^{1} \langle \nabla f( x+ \tau (y - x)), y - x \rangle d \tau $

I was wondering if anyone has a good explanation why this holds. I came across this in the page 17 of this paper (equations at the end of the page): $$ f(y) - f(x) = \int_{\tau = 0}^{1} \langle ...
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1answer
37 views

triangular inequality

If we write $||a+b||\leq||a||+||b||$ explicitly in $\mathbb{R}^n$ it is $\sqrt{\sum^n_1(a_i+b_i)^2}\leq \sqrt{\sum^n_1(a_i)^2}+\sqrt{\sum^n_1(b_i)^2}$ how can it be if ...
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2answers
25 views

Finding the basis for a subspace given the span of a set of vectors

Let U = span{$u_1, u_2, u_3$}, where $u_1 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} $, $u_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} $, $u_3 = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix} $, We ...
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1answer
69 views

“Hard” exercises on Linear Algebra and Analytic Geometry

I started lecturing this subject called "Linear Algebra and Analytic Geometry" and in the second day of class I was approached by an undergrad student, asking for referenced that would contain "hard" ...
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1answer
25 views

Understanding Composition Function (fg)(-1) for f(x)=x-3 & g(x)=x^2-8x+15?

Can someone help explain how to do the following composition function to me? (or at least get me started) Find the value of (fg)(-1) if ...
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2answers
43 views

Using the harmonic mean to determine the time to perform a task with varying manpower. A/K/A “Frank, Bob and Diane” [on hold]

Each year West Coast Shipping provides transportation to Pebble Beach Concours D'elegance. Our 3 drives Frank, Bob and Diane can deliver all the vehicles from the auction to West Coast Shipping ...
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0answers
10 views

Good video course or Maple worksheets which help to understand linear algebra? [on hold]

I need a good video course or Maple worksheets which help to understand linear algebra at a university level?
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2answers
35 views

I'm having a problem with this number and digits problem. What to do?

During the last election , the total number of votes recorded in the municipality of San Juan was 8600. Had one-third of Estrada’s supporters stayed away from the polls and one-half of Arroyo’s ...
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2answers
36 views

Is the maximum of the eigenvalues of any symmetric positive?

Let $A$ be a symmetric matrix having dimension $n \times n \;, \; \; n\geq 2$. If one wants to pick the maximum of its eigenvalues, will the value be positive? Suppose A was an adjacency matrix, ...
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1answer
25 views

Evaluating a limit involving the power of specially structured matrix

Let $k\times k$ right-stochastic matrix $A$ be defined as follows: $$A=\left[\begin{array}{cccccc} p & 0 & 0 & \cdots & 0 & 1-p\\ 1 & 0 & 0 & \cdots & 0 & 0 \\ ...