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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Real life applications of general vector spaces

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of ...
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0answers
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inequality with a positive matrix

Let $$ A=\left[ \begin{array}{cc} a & b\\ \overline{b} & c\\ \end{array} \right]$$ be a positive semi-definite positive of $M_2(\mathbb{C})$. How prove the inequality $ac \leq ...
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0answers
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Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
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votes
3answers
71 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
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1answer
10 views

eigenvalue and rank of a transformation

what i feel is that since the range of the linear transformation is strictly less than $n$ this implies that the transformation is not onto hence the null space contains a non trivial vector.but is ...
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0answers
8 views

Find orthogonal projection to x-y, x-z, and z-y, plane

In linear transformation from $R^3$ to $R^3$, how would you find the matrix of the linear transformations to do these projections?
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1answer
18 views

Tensor Product of Spaces has Basis of Tensor Products

I am given the following definition of the Tensor Product of spaces Given two vector spaces $V,W$ a vector space S is a tensor product of $V,W$ if there exists a map $M$ $$ M: V \times W ...
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1answer
19 views

Finding a Diagonal Matrix for a Linear Transformation

here is the problem: I am pretty stuck on this one. I thought that the formula for a projection was: wx/ww times w, which in turn forms a matrix [w1^2, w1w2], [w1w1, w2^2] * 1/ (w1^2 + w2^2), but ...
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1answer
34 views
+50

Idempotents which are not Mouray von neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Mourray Von neumann equivalent to $e^{*}$?
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0answers
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Proving Holder's inequality for Schatten norms

Sticking to the finite dimensional case, Holder's inequality for Schatten norms is given by $$\left\|AB\right\|_{S^1}\leq\left\|A\right\|_{S^p}\left\|B\right\|_{S^q}$$ for $A,B$ $n\times n$ ...
0
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1answer
23 views

Find the image of the transformation and write as a span of vectors.

Let $T(a,b)=(a+b,2a-b,3a)$ Find the image of $T$ (as a span of vectors). So I created the augmented matrix and got this: $A$= $\begin{bmatrix}1 & 1 & b_1\\2 & -1 & b_2\\ 3 &0 ...
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1answer
388 views

changing bases/rotating axes to find reflection across y=2x

Find the (exact) reflection of the vector v = (5, 1) across the line: y = 2x. Hint: A sketch of v and the line may suggest an approach. I found the matrix -3/5 6/5 4/5 2/5 which seems like it gives ...
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2answers
22 views

The definition of a subspace in linear algebra

I'm trying to learn linear algebra on my own but I am stuck on the definition of a linear subspace. Let's assume I want to find out if $S$ is a subspace of $\mathbb{R}^2$, where $ S = [X_1 , X_2] $ ...
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0answers
18 views

Writing the space of all possible solutions using a homogenous and particular solution

System of equations: $$\begin{align} 2w + 3x -2y +z &=-1 \\6w+ 10x \quad +6z&=14 \\3w +2.5x -15y -4.5z &= -35.5 \end{align}$$ Particular solution to the system of equations: $A$= ...
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0answers
10 views

A matrix multiplication problem

Suppose we have been given $2n^2$ vectors $a_1,\dots,a_{n^2}$ and $b_1,\dots,b_{n^2}$ each in $\Bbb Z^{n}$. Form an $n^2\times n^2$ matrix $M$ with $i$th row given by $a_i\otimes b_i$. What ...
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0answers
14 views

Determining if a function is linear, time invariant, both or not

I have the function $y(t)=t^2x(t-1)$ and I need to figure out if it is linear or not and time invariant or not. By the looks of it I guessed it to be not linear but the answer is linear but not time ...
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0answers
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Intersection of tensor product of vector spaces whose intersection is $\{0\}$ is trivial

Let $V$ and $W$ be subspaces of a finite-dimensional vector space $U$ such that $$V \cap W= \{0 \}.$$ Let $A$ be a second vector space (possibly infinite). Is it true that as subspaces of $A ...
0
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1answer
30 views

Area of a parallelogram with three points in $\mathbb{R}^{n}$: $(a,b, 0); (a, 0, b); (0, a, b)$

I have been requested to calculate the area of the parallelogram with three adjacent vertices: $(a,b, 0); (a, 0, b); (0, a, b). First, I have made this diagram: Then I proceed to calculate the two ...
0
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1answer
7 views

Additivity Implies Homogeneity of Rational Scalars

I did my best to search for this question on the site- but I did not find it. Here it is: If a function $f:\mathbb{R}^2\to\mathbb{R}$ satisfies $f(u+v)=f(u)+f(v)$ for all $u,v\in\mathbb{R}^2$, ...
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1answer
17 views

If a subspace of a finite-dimensional vector space. Then the subspace is finite dimensional?

I have difficulty in understanding the proof of this statement: Let W be a subspace of a finite-dimensional vector space V. Then W is finite dimensional. The proof goes like this. (Linear algebra, ...
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3answers
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Condition for right handed invertibility

Suppose that $A$ is an m by n matrix and is right invertible, such that there exists and an n by m matrix $B$ such that $AB = I_m.$ Prove that $m\leq n.$ I'm not really sure how go about this ...
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2answers
43 views

How can I show that for matrix $A$ , $A^t A $ is not equal to $ A A^t $ in general?

How can I show that for matrix $A$ , $A^t A \neq A A^t $ $A^t$ means the transpose of $A$. That is the entire question and I have no idea how to begin... please help!
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1answer
66 views

$e^{(A+B)} = e^Ae^Be^{[A,B]}$ for non commuting A and B?

For non commuting A and B, and the derivative of $[A,B] = 0$. Is it true that/how to prove that $e^{(A+B)} = e^Ae^Be^{[A,B]}$ If not, what is the expression according to Wikipedia's article on the ...
0
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2answers
33 views

Find the matrix representing T and Find the Image of T (as a span of vectors)

Let $T(a,b) = (a+b,2a-b,3a)$. a)Find the matrix representing $T$. b)Find the image of $T$ (as a span of vectors). So I found that $T$ is a linear transformation. Now would the matrix just be $A$= ...
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1answer
17 views

What are the $GL_n(F)$-orbits of a group action on the set of idempotent matrices?

Let $S= \{A \in M_{n \times n}(F):A^2=A\}$ (set of idempotent matrices). The general linear group $G=GL_n(F)$ acts on $S$ by $A.g=gAg^{-1}$ (conjugation). I'm having trouble visualizing the ...
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2answers
48 views

Subspace of linear transforms from V to V

Suppose V is finite-dimensional and the $\mathscr{E}$ is a subspace of $\mathscr{L}(V)$ such the $ST\in \mathscr{E}$ and $TS \in \mathscr{E}$ for all $S \in \mathscr{L}(V)$ and all $T \in ...
0
votes
1answer
14 views

Range of sum of vector space

Let $S,T$ be elements of $L(V,W).$ Show that the range$(S +T)$ is a subspace of range$(S)$ + range$(T)$. I tried applying the definition of range, but I wasn't sure how to proceed after that.
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2answers
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Find $\phi, \psi: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, each nilpotent of order 2, whose composition is idempotent

$\phi: V \rightarrow V$ is nilpotent of order 2 if $\phi \phi$ is the zero endomorphism. Now composition of two such endomorphisms need not be nilpotent of order 2. Find $\phi, \psi: \mathbb{R}^2 ...
2
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1answer
28 views

why does matlab give me a negative number?

I have the following problem A steel company has four different types of scrap metal (called Typ-1 to Typ-4) with the following compositions per unit of volume They need to determine the volumes ...
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1answer
14 views

Image and Kernel of a Projection of One Line onto Another

The question is: Let T be the projection along a line L1 onto a line L2. Describe the the image and the kernel of T geometrically. I understand that the image should be the Projection of L1 onto L2. ...
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1answer
45 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
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1answer
18 views

Proving a norm is lipschitz

Let $M\in\mathbb{R}^{n\times n}$. Define the function $f\colon\mathbb{R}^n\to\mathbb{R}$ by $f(x)=\Vert Mx\Vert$. Show that $f$ is Lipschitz. Let $x,y\in\mathbb{R}^n$, then we want to find a ...
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2answers
32 views

How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans?

Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$ Apparently $\dim W_1=\dim W_2=2$. For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as ...
0
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1answer
35 views

Show that $P_n$ is an $(n+1)$-dimensional subspace of the vector space of all real polynomials [duplicate]

Show that $P_n$ = {polynomials with real coefficients of degree $\leq$ n} is an ($n+1$)-dimensional subspace of the infinite-dimensional vector space of all real polynomials I know that $P_n$ is ...
0
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1answer
18 views

Show that $Im T$ and $U/Nuc T$ are isomorphic for a linear transformation $T: U \longrightarrow V$

Show that $Im$ $T$ and $U/Nuc$ $T$ are isomorphic for a linear transformation $T: U \longrightarrow V$ Hi guys, I know how to show this for vectorial spaces with finite dimension, but I don't have ...
0
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1answer
33 views

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$ with $X_i$'s $ \in \mathbb{R}$ Just from computing $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$ I am guessing the general formula is: $(x_1 + \cdots + ...
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0answers
7 views

Derivation of variance of a linearly transformed vector

I am trying to derive the variance of a linearly transformed vector. A result was given here. $$ \mathbf{y} = X \, \mathbf{b} $$ $$ \mathbf{b} \sim \mathcal{N}( \mathbf{0}, \sigma^2 I) $$ If we say ...
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1answer
403 views

Simple Probability Matrix

Question: Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will ...
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1answer
11 views

What is the relation between the cokernel with the kernel of the dual map of a linear transformation?

I am studying linear algebra and I am in front on questions like: What is the relation between the kernel of a linear map and the cokernel of the dual map? What is the relation between theese objects ...
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0answers
21 views

Determinant of an elementary matrix

I read a very slick proof of determinant properties, in this case of the fact $\det A = \det A^T$, which says in one place It suffices to notice that for any elementary matrix $M$ we have $\det M ...
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0answers
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Understanding a proof of RREF uniqueness

Base Case $(n = 1)$: Suppose $A$ has only one column. If $A$ is the all zero matrix, it is row equivalent only to itself and is in reduced row echelon form. Every nonzero matrix with one column has ...
2
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1answer
18 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
2
votes
3answers
68 views

Why do I have to show this subspace is an invariant subspace?

Consider a vector space $V \cong \mathbb{R}^n$ with an operator $I \in O(n)$ satisfying the property $I^2 = -Id_{V}$. See Linear Complex Structure for context. I want to show that $V$ has real ...
3
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1answer
379 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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1answer
35 views

Define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine ker$\phi$ and im$\phi$

Let {$e_1,e_2,e_3$} be the standard basis for $\mathbb{R}^3$ and define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine the subspaces ...
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1answer
21 views

How do I get a solution set equals to a sub space?

I've four vectors that spans the $\mathbb{R}^4$ sub-space $W_1$: $\alpha_1 = \{-1,0,1,2 \}$, $\alpha_2 = \{3,4,-2,5 \}$, $\alpha_3 = \{0,4,1,11 \}$, $\alpha_4 = \{1,4,0,9 \}$ And I'm asked to ...
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1answer
46 views

MATLab help involving Linear Algebra.

I am trying to understand what is happening here.. A = floor(10*rand(3)); b = sum(A')'; z = ones(3,1); If I solve the system $Ax = b$, the solution is identical ...
0
votes
1answer
26 views

Dimension of the image of a matrix

So the question asks: Verify if the image of the linear map $T : \mathbb{R}^6 \to \mathbb{R}^3$ given by left multiplication by A= $$\begin{bmatrix}6 & 0 &2 & 2& 3& 4\\0 & -1 ...
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votes
1answer
20 views

How to construct an isomorphism between $ \ker g^{\ast}$ and $~coker~ g$?

Let $g: L \to M$ a linear transforming. $M, L$ finite dimensional. $g^{\ast} : M^{\ast} \to L^{\ast}$ How do I construct an isomorphism between $ \ker g^{\ast}$ and $coker~ g$? I really don't know ...
1
vote
1answer
22 views

choosing a square matrix to have a product with one 1 und other 0's

Let $A$ be a $m\times n$ real matrix with maximal rank. Let $i\in\{1,\dots,m\}$, $j\in\{1,\dots,n\}$. I'm curious if it is possible (for any choice of $i,j$) to find a square matrix $B$ such that ...