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 the distance between line and point R3

I would like to know how I can find the distance between the line and point in R3 the equations of line: $$ \left\{ \begin{array}{c} 2x+y+z=2 \\ 3x+4y-z=3 \end{array} \right. $$ and point (3, ...
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29 views

What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
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1answer
23 views

Proving a transformation is a linear transformation

If I have a transformation $T:V \to V$ given by $T(f(x)) = x*f(x)$. To prove this would I just show that $T(a*f(x_1) + b*f(x_2)) = a*T(f(x_1)) + b*T(f(x_2))$ to show addition and scalar multiplication ...
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2answers
30 views

to find the determinant of a matrix given the roots of an equation

Given an equation $x^2 +2x +2=0$ and $a$ and $b$ is the root of this equation. Find the determinant of This matrix \begin{bmatrix} a & 1 & 0\\ 0 & b & 1\\ 1 & b & 0 ...
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0answers
46 views

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices? [duplicate]

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices or having to choose a particular base ? Such a proof should probably use a non matricial definition of traces. One ...
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0answers
17 views

Expectation of matrix product

Suppose we have a random matrix $M \in \mathbb{R}^{n\times m}$ such that $\text{E}[M] = 0$ and $\text{E}[M M^\top] = \Sigma$. How does one compute $\text{E}[M^\top M]$?
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4answers
35 views

How are standard basis of polynomials linearly independent?

Consider the set $\{1,z,z^2,...z^m\}$. As this is the standard basis for a vector space of polynomials, the list should span the space and also be linearly independent. One problem I'm having though ...
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0answers
13 views

Prove that enumerable set of complex exponentials is linear independent

Define $f_j(p) = e^{i u_j \cdot p}$ for $j=1,2,3,...$, $u_j, p \in \mathbb{C}^N$, $i = \sqrt{-1}$ and $\cdot$ is the scalar product. I need help to prove that the set $\{f_j : j=1,2,...\}$ is linearly ...
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0answers
9 views

Has any system a minimal subsystem [on hold]

Has any system a minimal subsystem? I can't really grasp the concept, could someone maybe provide an example?
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1answer
81 views

Show that A and B commute

Let $A$ be a Hermitian matrix. Suppose there exists a matrix $B$ such that $A^3B = BA^3$. Show that $AB = BA$. I was trying to use the fact that since $A$ is Hermitian, there exists a unitary ...
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0answers
17 views

Formula to calculate angle on a fan or semicircle

How do I calculate the angle shown in the picture given the height, width, and the arc deduction of $2$? I had applied the Right Triangles formula to calculate the hypotenuse: $h^2 = a^2 + ...
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1answer
33 views

Is it a subspace or not?

Is $$\mathscr{S_1}=\lbrace(a,b,c)\in\mathbb{R}^3:a^3=b^3\rbrace$$ subspace of R^3? my try:: $a^3=b^3\iff a=b$ in $\mathbb{R}$. So $\mathscr{S_1}=\lbrace(a,a,c)\in\mathbb{R}^3\rbrace$ this is a plane ...
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1answer
26 views

Linear Transformation on R^6 [on hold]

I am stuck on this linear transformation problem.Thanks for the help!
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2answers
32 views

Linear Algebra. Closed unit ball. Prove. [on hold]

Hi, dear all, how can I prove that? Please help me. I need your help very much!
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0answers
39 views

Give me an idea [on hold]

We've got an infinite number of cards, each of them having a positive integer written on it.Prove that however we choose 2015 cards, having the sum of the numbers written on them 4028, we can divide ...
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vote
3answers
30 views

Proving a transformation is not a linear transformation

I'm asked to prove if a transformation is linear or not. In the vector field $V=\{f(x)\colon \mathbb{R} \to\mathbb{R}\}$, so the transformation is $T\colon V \to V$ given by $T(f(x)) = (xf(x))+1$. I ...
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2answers
38 views

Finding Eigenvalues of given linear operator

Find the eigenvalues and the eigenvectors of the linear operator $T:C^\infty(0, 1)\to C^\infty(0, 1)$ $T(f)(x) = \frac{f'(x)}{x}, x \in (0,1) $ Using the definition : $TF = \lambda F \iff ...
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2answers
16 views

Is the rank of a matrix unaffected by congruence transformations?

Sorry for the easy question but if I have a square matrix $A$ over $\mathbb C$ then is its rank invariant under a congruence transformation $A \mapsto P^t AP$ ? What's the easiest way to see this? ...
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2answers
20 views

Linear operator matrix

Why if a linear operator $\phi$ is an isomorphism and we multiply the matrices of $\phi$(x).$\phi^{-1}(x)$ in any basis, the result is E the matrix that has 1 on the diagonal and everything else is 0. ...
2
votes
2answers
24 views

Skew symmetric 4x4 matrix of full-rank

I have come across the fact that a 4x4 skew-symmmetric matrix of full-rank is equivalent to \begin{pmatrix} 0 &\theta_1& 0 &0 \\ -\theta_1& 0 &0 &0 \\ 0& 0&0 & ...
2
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2answers
22 views

Linear transformation with special properties

how should I do that please (I had this in my test yesterday)? Linear transformation $f:\mathbf{R}^{10} \to \mathbf{R}^7$ has an attribute that every vector $\mathbf{v}$ for which is true that ...
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1answer
17 views

invariant subspace under some conditions

This came out at my linear algebra exam and I was not able to solve it. Let $f\colon \mathbb{R^{3}} \rightarrow \mathbb{R^{3}}$ be a linear transformation such that $\langle f(u),f(v) \rangle = ...
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0answers
7 views

Need to prove a property using super modularity and convexity

I have a function f(x,y) that is convex on both x and y separately (but maybe not a joint convex function). It is also super modular. Assume that we have ordered sets of x and y as (x1>x2,x3>x4) and ...
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1answer
21 views

Can a comparison network obtain all the n! permutations of a vector?

I want to permute a vector using comparison networks. This is the only method I have at my disposal. My original idea is to use a sorting network like Batcher or Bitonic. Basically I place my vector ...
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2answers
18 views

Definition of an image of a linear transformation

I have the following definition of an image of a linear transformation, but I think that there's a mistake in the size of the field (confusion between m and ...
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0answers
19 views

Find the image of a vector

I have an endomorphism and I have to study the image of a vector $v$. How can I do this? Can you please give an example? I know how to calculate the inverse image but I have many doubts on the ...
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0answers
19 views

Is there a relation to determine condition (positive or negative definite) of C, if C = A+B and A, B are positive and negative definite?

I have a question: Matrix A and B are positive and negative definite, respectively. Is there a relation to determine whether C is positive or negative definite, if C = A+B?
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3answers
51 views

If $X + X^T$ is positive definite, is $X^{-1} + X^{-T}$ also positive definite?

Is it true or is there a counterexample?
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0answers
7 views

$\frac{4}{3}(1,-1,0)^T+\textrm{vect} \left((-1,1,-1)^T\right)$

Here I have a passage on the conclusion, but I do not understand it. the conclusion say : $f$ is the oriented axis of screw.. how I can from $\frac{4}{3} \begin{pmatrix}1\\1\\0 ...
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1answer
31 views

How to bound the biggest eigenvalue of $\sum_{i=1}^{n}x_ix_i^T$?

My question is to bound the biggest eigenvalue of $A=\sum_{i=1}^{n}x_ix_i^T$, where $x_i\in\mathbb{R}^d$ is a column vector. My idea is, to bound the biggest eigenvalue of $A$, i.e. $\|A\|_2$. I can ...
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0answers
18 views

What is a “supplementary subspace”?

Let $Q$ be a quadratic form of vector space $V$ over a field $k$ with characteristic $\neq 2 $, $V^{0}$ be its orthogonal complement. If $U$ is a supplementary subspace of $V^0$ in $V$, then $V = ...
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0answers
19 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
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3answers
42 views

Show that $\operatorname{span}(\operatorname{span}\{\vec{x},\vec{y}\}) = \operatorname{span}\{\vec{x},\vec{y}\}$

If I am given fixed vectors $\vec{x},\vec{y}\in \Bbb R^n$, how can I show that $\operatorname{span}(\operatorname{span}\{\vec{x},\vec{y}\}) = \operatorname{span}\{\vec{x},\vec{y}\}$? I am a little ...
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0answers
11 views

extract the vector X in an equation

For the following question I need to extract the X from the equation $A = \left[ {B_{0,0}^TX, \ldots ,B_{0,M - 1}^TX, \ldots ,B_{N - 1,0}^TX, \ldots ,B_{N - 1,M - 1}^TX} \right]$ where $X$ and ...
1
vote
2answers
26 views

Algebra QF $f(x)=x^2-18x-4$

I have the problem to find the zero of the function: $$f(x)=x^2-18x-4$$ I have it mostly worked out as $a=1, b= -18, c= -4$ worked out I have: $$\frac{18\pm \sqrt{340}}{2}$$ I know the answers are ...
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1answer
9 views

Rank of matrix in relation to number of rows and columns

From my linear algebra text: “Suppose that rank $A = r$, where $A$ is a matrix with $m$ rows and $n$ columns. Then $r < m$ because the leading 1s lie in different rows, and $r < n$ ...
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1answer
24 views

Is every finite dimensional linear space a banach space [on hold]

Is every finite dimensional linear space a Banach Space? Is every finite dimensional linear space a Hilbert Space?
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2answers
40 views

The purpose of LU Decomposition

I was curious if anyone could help me understand why an LU decomposition is useful from a theoretical or computational standpoint. It seems to me that it is just a way to teach students the basics of ...
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1answer
19 views

Graphing linear, affine, and convex combinations

For the vectors (2, 1) and (1, 3), how would I graph each of the three combinations? Here are my thoughts (sorry might be totally wrong): linear - plane connecting the two points affine - infinite ...
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1answer
19 views

Given a square matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues?

Given a matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues? If instead the matrix had its nonzero entry component at ...
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0answers
21 views

How to determine if a set represents a line, plane or hyperplane?

How do you approach a question that gives you a set and asked to determine if it represents a line, plane or hyperplane? The Question: https://www.dropbox.com/s/0gscqur18kqg3ma/SpanningQuestion.PNG ...
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0answers
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How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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1answer
27 views

Using absolute value to solve an equation.

I am required to have two answers to this problem: A 'Larger Number' and a 'Smaller Number'. I know that I have half of the question correct (The 'Larger Number' answer). But I do not know how to ...
0
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2answers
50 views

Matrix invertible iff det(matrix)$\neq 0$?

When we want to find the inverse of the matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ we're searching for a matrix $$\begin{bmatrix}x & y \\ z & w\end{bmatrix}$$ such ...
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0answers
33 views

prove the subspace is equal to L1∩L2

Let $L1,L2$ be subspaces of $V$ and $dim(L1+L2)=1+L1 \cap L2$ Show that $L1 \subseteq L2$ or $L2 \subseteq L1$ and $L1+L2= L1$ or $L2$ it's has something to do with if $L1,L2 \ne L1 \cap L2$ ...
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0answers
23 views

Find value of x from the span of vectors. [on hold]

Find 'x' such that the vector is the span of set of vectors:
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1answer
27 views

Relationship of basis vectors of the complex plane

I am working on learning more about the connection of complex numbers and rotations in the context of rational geometry. Thanks ahead of time for any corrections on my best assertions. Let $B$ ...
1
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1answer
19 views

Triangularization of a matrix.

so I need to find an invertible matrix $P$ such that $P^{-1}AP$ is upper-triangular, where $$A = \begin{bmatrix} 4 & -1 \\ 9 & -2 \end{bmatrix}$$ So I found that the eigenvalue is $1$ which ...
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0answers
22 views

By using the properties of determinants prove that [on hold]

By using the properties of determinants, prove that:
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1answer
28 views

About Lines and Planes in Linear Algebra

The set of linear combinations $cX$ is said to span a line. How can I see that there's correspondence between $cX$ and the equation of a line $ax + by = 0$? Besides, the aforementioned equation looks ...