Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Algebraic characterization of a union of two affine subspaces

Is it a simple algebraic characterization of affine hull of the union of an affine set $A$ in a linear space and a point $x$ not lying in this hyperspace? I thought that it is $$\{(1-t)a+tx: t\in ...
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Eigenvalues of a nilpotent matrix can only be $0$ [duplicate]

Prove that the eigenvalues for a square Nilpotent matrix A can only be $0$. Definition of nilpotent A $^n$=$0$ n is a positive whole integer
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55 views

Under what assumptions is it correct to say “a matrix is diagonalizable if and only if its eigenvalues are real”?

A $2\times 2$ matrix is diagonalizable if and only if its eigenvalues are real. Which statement is most correct: The proposition is true only if the eigenvalues are all greater than zero. The ...
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Finding polynomial generators in a subspace

$S$ is a subspace $S= \{p\in P_3|~\text{$i\in\Bbb C$ is root of $p$}\}$. So the question at hand is how do you find the system of generators for the subspace knowing that $x$ is $p$'s divisor? ...
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Distance between point and translated subspace

Given $(m - 1)$-dimensional subspace of $n$-dimensional space ($m \leq n$), that is defined by a set of $m$ its (linearly independent) points. How to compute the distance between separate point $p_0$ ...
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Converting nth order ODE with RHS into system of 1st order ODEs

I looked at these two questions, but they weren't directly relevant to my specific question: How to reduce higher order linear ODE to a system of first order ODE? Express differential equations as ...
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24 views

Under what condition on a set $S$ does $f$ exist such that $f \cdot S_k$ has the same value for all $S_k\in S$?

this is my first time using this site so I apologize if I'm unclear or using poor convention. I'm working on a problem with wireless power transfer, which long story short involves a set of transfer ...
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Problem of Closed linear transformation in Normed spaces [duplicate]

Let $X$ a normed space and let $A$ and $B$ be linear transformations such that $$X\subset D_A\rightarrow^{A} X \ \ \text{and} \ \ X\subset D_B\rightarrow^{B} X.$$ If $A$ and $B$ are closed, does it ...
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26 views

Prove there exists a sequence of bounded numbers $(v_1,v_2,..)$ such that $\sum_{n=1}^\infty a_nv_n = f(x)$?

Let $(X,\|\cdot\|)=(l^1,\|\cdot\|_1)$ and let $f$ be a bounded linear functional. Prove there exists a bounded sequence $(v_1,v_2,...)$ of real numbers such that: ...
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Which of the following subsets of P2 are subspaces of P2?

{p in P2: p(0) > p(1)} {p in P2: p(3) = p(4)} {p in P2: p'(3) = 4p(7)} I understand that generally we need to check to make sure that these are closed under addition and scalar multiplication, and ...
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linear algebra, space vectors

I am having difficulties in proving whether a particular E is space for example IR, so I would like if you could do this exercise. Determine whether IR ^ 2, with the operations described, is a real ...
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linear algebra, space vetors

I am having difficulties in proving whether a particular E is space for example IR, so I would like if you could do this exercise. Determine whether IR ^ 2, with the operations described, is a real ...
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13 views

Find the orthogonal projection P on L.

Let $L=<3e_3+2e_1,e_5>$ and $x=(2,1,3,2,-6,8,2,1,0,0,0,...)$. Find $\|Px\|$ where P is the orthogonal projection on L (Do not forget that the formula for Px orthogonal vectors need length 1) I ...
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Find an orthogonal matrix that achieves a given vectorial transformation

Given a vector $\vec a\in\mathbb R^n$ and another $\alpha=(\|\vec a\|,0,\dots,0)$, how could I define an orthogonal matrix $M$ such that $M\vec a=\alpha$ and $M^{-1}=M^t$? For $\mathbb R^2$ I tried to ...
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How prove this rank identity $r(A)=r(B)$

let $A_{n\to n},B_{n\to n}$ matrix,and such $$A^2=20142014A,B^2=20142014B,$$ and $20142014I-A-B$ is invertible, show that $$\rm{rank{A}}=\rm{rank{B}}$$ we know $$A^2-20142014A=0$$ then ...
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How to find the characteristic polynomial of this transformation?

Let V be a finite-dimensional inner product space, and let W ⊂ V be a subspace. Let T : V → V be the linear transformation “orthogonal projection onto W”: T(x) = ProjW x. Show that T is ...
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$p(x)$ divides the minimal polynomial iff $\exists v\ne 0: p(T)(v)=0$

Let $V$, a finite dimensional space. Let $T:V\to V$ a linear transformation. Show that $p(x)$, an irreducible polynomial divides $m_T$ (The minimal polynomial of $T$) iff there is a $V\ni v \ne 0$ ...
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Linear Transformation of standard matrix [on hold]

Please help me for solving following problem a) find the standard matrix $A$ for the linear transformation $T$, b) Use $A$ to find the image of the vector $v$. $T$ is the counterclockwise rotation ...
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Give an example of a linear subspace $L$ of $H=l^2$ such that there exists no $y\in L$ such that $\|x-y\|=$dist($L,x$)?

I know that a subspace that is closed and convex must have a unique y in L such that it is true and that if it is closed presumably then you can have many y. So I am looking for an L which is open in ...
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Finding complex eigenvectors of $n \times n$ matrix, $n\geq 3$

An example: $$ \begin{pmatrix} 1 & 2 & 0 \\ 2 & -3 & 4 \\ 4 & -8 & 7 \\ \end{pmatrix} $$ Has eigenvalues $3$, $1+2i$, $1-2i$. How ...
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23 views

Gram-Schmidt process in function subspace

I have a function space $\mathcal {F}([-1,1],\mathbb R)$ and the subspace $\mathcal{P_2}:=$ $(x\mapsto a_o+a_1x+a_2x^2| a_0,a_1,a_2 \in \mathbb R )$ for all polynomials with degree $\le2$. In this ...
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If $T^k = Id$ for $k\ge 1$ then $T$ is diagonalizable [duplicate]

Let $V$ a finite dimension space over $\mathbb{C}$ and $T:V\to V$, a linear transformation such that $T^k = Id$ for $k\ge 1$. Prove that $T$ is diagonalizable. I'd be glad for an hint. How do I ...
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Is my proof correct that function is a bijection iff matrix is invertible?

For given $B\in \mathbb{C}^{n\times n}$ let's define a function $$f: \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$$ so that $$f(A) = B^HAB$$ I have to prove that f is a bijection iff B is ...
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Question on Logic in translation

let $P,P'$ two affine subspace of $R^{3}$ have we equality between this two statement $$\exists\ u_{0}\in R^{3}\ \mbox{such that } t_{u_0}(P)=P'$$ $$\exists B,A\in PP' \mbox{such that } ...
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is this kind of symmetric matrix invertible?

Give a matrix $A=\begin{bmatrix}M&B\\ B^T&0\end{bmatrix}$, where $M\in\mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}, (m<n)$. If we know that $rank(B)=m$ and for any $v\neq 0$ and ...
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polynomial algebra and multiplications of its elements

According to the definition of the polynomial algebras $A(n)$ and $A(n,m)$ for n∈N and $m∈N^n$, if F be field GF(2) and $X_1,...,X_n$ be n pairwise commuting indeterminates over GF(2), then for a∈N we ...
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Generalized Eigenvector Problem

I am reviewing a paper in which the solution to $$x =\max_\bf{v}\frac{\alpha \bf{v}^\dagger \bf{h}\bf{h}^\dagger \bf{v}}{\beta+\gamma\bf{v}^\dagger \bf{D} ...
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Can a matrix transformation ever make a linearly dependent matrix linearly independent?

I'm curious. Can ANY matrix transformation make some matrix with its columns linearly independent, or with an empty kernel, linearly independent? For example, if A is a linearly dependent matrix, and ...
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Vector question involving an operator!

So, here's the problem: An operator H capable of operating on vector x, is defined in terms of a given vector a by: H x=(a * x) where $*$ representes vector product Given that ...
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Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
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Solving a linear Equation!

I have the equation i.e. $4xy=x$ and I have its two solutions i.e. $x=x$ and $y=1/4$ and $x=0$ and then $y=1/4$. I am puzzled that whether it is a equation merely defining the line that is, $y=1/4$ or ...
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Given that A is an nxn symmetric matrix, show that $(Av)\cdot w=v\cdot (Aw) $

Here v and w are column vectors in $R^n$. What are the properties that I need to use to show this is true? Edit: Just needed tranpose, so we have $(Av)w=(Av)^Tw=v^tA^tw=v^TAw=v^T(Aw)=v\cdot(Aw) $
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Confusion about change of basis matrix

This video here seems to suggest that if a vector $v = (c_1, \dots, c_n)$ is given with coordinates in some basis $b_1, \dots, b_n$ and $B$ is the matrix with columns $b_1, \dots, b_n$ then $Bv$ is ...
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Can anyone help me with “rotation matrix” and “Image of matrix”?

If A is a 3 by 3 matrix which gives a rotation about some line through the origin in R^3 , then columns of A form a basis of R^3 For any matrix A, the image of A^7 is contained in the image of A ...
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How does pointwise multiplication of two matrices affect their eigenvectors?

More specifically, suppose I have a known matrix $X\in\mathbb{R}^{d\times n}$ and an unkown vector $\alpha \in \mathbb{R}^n$. What can be said about the eigenvectors of $\alpha\alpha^T \odot X^T X$ ...
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Linear transformation, matrix and basis

Let $T$ be a linear transformation on $R^3$ whose matrix relative to the basis $\{\beta_1,\beta_2,\beta_3\}$ is\begin{pmatrix}1&2&-1\\2&0&2\\1&-2&3\end{pmatrix}. Show that ...
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On the importance of order for bases in finite dimensional vector spaces

I am reading Tapp's Introduction to Matrix Groups for Undergraduates and he writes: Let $V$ be an $n$-dimensional (left) vector space over $\mathbb K$. Then $V$ is isomorphic to $\mathbb K^n$. In ...
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1answer
41 views

Show the comutativity of matrices

Let $\Omega$ be the set of some $n\times n$ matrices with the property $$A,B\in \Omega\Rightarrow AB\in\Omega, (AB)^3=BA.$$ Show that $AB=BA,\forall\ A,B,\in \Omega$. What I could show now is ...
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Generic rotation to remove Quadratic Cross-product

Show that if $b\neq 0$, then the cross-product term can be eliminated from the quadratic $ax^2 + 2bxy + cy^2$ by rotating the coordinate axes through an angle $\theta$ that satisfies the equation $$ ...
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Finding $n$ scalars such that $\det{(cI-A)}=0$ without eigenvalues

My problem is this Let $A$ be an $n\times n$ matrix over $\mathbb{F}$. Prove there are at most $n$ distinct scalars $c\in\mathbb{F}$ such that $\det{(cI-A)}=0.$ I know that the determinant is ...
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Finding equations for a subspace in non-standard operations

There is a Vector space $\Bbb V$ with these non-standard operations (with component to component multiplication) [A]$_i$$_j$ $\oplus$ [B]$_i$$_j$ =[A]$_i$$_j$ . [B]$_i$$_j$ and ...
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I didn't figure out how the result in part (i) can help in (ii). Anyone has any idea??

The determinant turns out to be -3 in part (i) How can this help in showing that the 4 vectors in the end are linearly independent?
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Can we specify all row equivalent matrices of a given matrix?

Say we have a RREF matrix like $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}$$ From this matrix, is there some way of specifying ...
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Mutually orthogonal set of vectors

Show that the standard basis: $$..$$ $\mathscr{B} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0\\ 1 \\ 0 ...
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Linear Algebra: Properties of the Determinant

On a recent exam, I was given the following problem: Suppose that $\det(A) = -3$, $\det(A + I) = 2$, and $\det(A + 2I) = 5$. What is $\det(A^4 + 3A^3 + 2A^2)$? I just don't see how the ...
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1answer
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Book suggestion to prepare the grounds for studying functional Analysis

Hi guys I have 2 month semester break in February and March and I am planning to take a course on functional analysis in 4 months. I have taken a very elementary course on Linear Algebra(Gilbert ...
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Prove that $T,S$ are simultaneously diagonalizable iff $TS=ST$. [duplicate]

Definition: We say that $S,T$ are simultaneously diagonalizable if there's a basis, $B$ which composed by eigen-vectores of both $T$ and $S$ Show that $S,T$ are simultaneously diagonalizable iff ...
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Determinant of exact sequence

Let $0 \to A \to B \to C \to 0$ be an exact sequence of vector spaces. I want to show that I have a canonical isomorphism $$\text{det}(B)= \text{det}(A) \otimes \text{det}(C).$$ Here, "det" refers ...
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1answer
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Ranks of a matrix and determinants of its minors

Given a $n\times n$ matrix M. Prove that M has rank less than $k$ if and only if all of the determinants of its $k\times k$ minors are $0$. My progress: I have thought about this problem for a while, ...
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Can anyone help me with these true and false questions about linear algebra?

1 A system of real linear equations can have exactly two solutions. 2 If U and W are subspace of V, V=U+W (Finite dimension), then dimV is less than or equal to dimU+dimW 3.Every inner product ...