Tagged Questions

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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1answer
12 views

Find the basis of set given by matrices

In linear space of matrix $2\times 3$ over $C$ we have subspace generated by: $ A= \{{\left[\begin{array}{ccc}i&i&i\\i&0&1\end{array}\right]}$ ...
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1answer
35 views

Help with Gram-Schmidt problem

I'm supposed to show that the Gram-Schmidt process: $\textbf{a}_j = \left\{ \begin{array}{lr} \textbf{d}_j, \;\;\textbf{if} \;\;\lambda_j = 0\\ \sum_{i=j}^n \lambda_i\textbf{d}_i ...
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0answers
16 views

minimal polynomial of $T$-invariant subspaces

I am stuck on this problem Problem: Let $V$ be a finite dimensional vector space and $T$ a linear transformation from $V$ to $V$. $W$ is a non-trivial $T$-invariant subspace let $S$ be the induced ...
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0answers
17 views

find the standard matrix for linear transformation

Find the standard matrix $T=T_2 \circ T_1$, where $$T_1:\Bbb{R}^3 \to \Bbb{R}^3, \qquad T_1(x,y,z)=(x+2y, y-z, -2x+y+2z)$$ and $$T_2:\Bbb{R}^3 \to \Bbb{R}^3, \qquad T_2(x,y,z)=(y+z,x+z, 2y-2z)$$
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0answers
24 views

Orthogonal diagonalization of a symmetric matrix

Find an orthogonal matrix $P$ that diagonalizes $$\begin{pmatrix}-1 &4 &-2\\ -3& 4 &0\\ -3 &1& 3\end{pmatrix}.$$ My eigenvalues are 1 , 2 and 3 but my $P$ while verifying is ...
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1answer
29 views

Eigenvalues and eigenvector of symmetric matrix

Compute eigenvalues and eigenvectors of the following matrix: $ \begin{pmatrix} 11 & 4 & 14 & \\ 4 & -1 & 10 & \\ 14 & 10 & 8 & \\ \end{pmatrix} $ 1.One ...
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0answers
14 views

Determine the possible eigenvalue of a linear operator.

Let $F:V \rightarrow V$ be a linear operator such that $$F(F (\vec{x})) = F(\vec{x}) + 2\vec{x} $$ $\forall$ $x \in V$. Determine the possible eigenvalues. Prove it. Here is my solution. I cannot ...
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1answer
13 views

Linear maps, inverses and associated matrices?

This is likely a very simple question but if we have a linear map $f$ with an associated matrix $A$ is it a necessary and sufficient condition that for $f$ to have an inverse then $A$ must also have ...
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0answers
16 views

What is the multiplicity of the largest eigenvalue of a graph?

The Laplacian of a graph is a symmetric positive semi-definite matrix and hence has all real eigenvalues. Is there any characterization for the multiplicity of the largest Laplacian (and/or Adjacency ...
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2answers
13 views

Eigenvector system with matrix on right side of equality

I am reading a math paper which describes the following system as an eigenvector system. $(D − W )v = λDv$ My linear algebra skills are rusty, but I thought an eigenvector system must be of this ...
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1answer
22 views

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?
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0answers
18 views

Can anyone help me with this linear differential question?

Given two constant-coefficient operators A and B whose characteristic polynomials have no zeros in common. Let C=AB first part of question is "Prove that every solution of the differential equation ...
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4answers
46 views

Prove determinant is zero

If $M = \begin{vmatrix} 1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b \\ \end{vmatrix}$ Show that M = 0 WITHOUT expanding the determinant. I ...
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0answers
12 views

$\phi:\mathbb{R}^2\to\mathbb{C}$,$\phi(x,y)=x+iy=z$,$F=\phi^{-1}f\phi$

$\phi:\mathbb{R}^2\to\mathbb{C}$ be a map $\phi(x,y)=x+iy=z$, let $f:\mathbb{C}\to\mathbb{C}$ be the function $f(z)=z^2$ and $F=\phi^{-1}f\phi$ then I need to say which of the following are correct. ...
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0answers
18 views

Problem in kernel of dual spaces. Show $(\ker{p(T)})^*$=$\ker{P(T^*)}$ [on hold]

So I have a finite dimensional vector space $V$ and and $T$ an endomorphism. Define $p(T)$ to be its minimal polynomial which is irreducible.I am wondering if the following holds: ...
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1answer
19 views

$T$-invariant subspace and minimal polynomial

This is the problem that I am stuck on. Problem: Let $V$ be a finite dimensional vector space and $T: V\rightarrow V$ be a linear transformation. Suppose ...
1
vote
1answer
21 views

Continuity of $f(x)=(xI-A)^{-1}$?

Let $A\in \mathbb{C}^{n\times n}$ and $I_n$ be an identity matrix. If $z\in \mathbb{C}$ is not a eigenvalue of $A$, then $f(x)=(xI-A)^{-1}$ is a continuous function at $z$. Is that correct?
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1answer
30 views

why is 1-lamba an eigenvalue of identity matrix -A?

someone posed this problem to me and it may be lack of sleep but i can't really figure it out. he said it was an easy problem too. ok so i have tried just assuming A is a 2x2 matrix so the ...
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0answers
18 views

Proving a theorem a about matrix derivations

Ok, so Im doing some research and I have to understand the following theorem. The theorem states: Let $h$ be a derivation on $Tn(R)$ with $h(e_{ij})=0,\,\, 1\le i \le j \le n$. Then $h=\bar\delta$ ...
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0answers
15 views

Is the product of (modified) adjacency matrices of two matchings necessarily symmetric?

Consider $n$ vertices, and two (not necessarily perfect) matchings $M_1$ and $M_2$. With the following definition of a (modified) adjacency matrix of a matching, can we claim that $A(M_1)\cdot A(M_2)$ ...
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1answer
30 views

Do I leave the 0 vector in my transition matrix?

The $2\times 2$ matrices of the form: $$ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & 0 \end{bmatrix} $$ where the entries $a_{ij}$ are all real numbers, form a subspace of the vector ...
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0answers
8 views

Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
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3answers
46 views

What is the kernel? [on hold]

Let $P_3$ denote the real vector space of polynomial functions of degree up to 3, i.e., $P_3 = \{ a_3x^3 + a_2x^2 + a_1x + a_0 \mid a_i \in R\}$. Consider the linear transformation $D : P_3 \to P_3$ ...
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3answers
21 views

Canonical isomorphism between $V$ vector space and its second dual $V^{\circ \circ}$

I came a across this when I was reading some book. It says let $V$ a finite dimensional vector space of some field and there is a canonical isomorphism $\phi$ between $V$ and $V^{\circ \circ}$ but ...
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2answers
23 views

Subspace of P_2? [duplicate]

Is the set of the polynomials in the form $cx^2+dx+e$ with $c+d+e=0$ a subspace of $P_2$? Why? Is there a zero component in this if $c=d=e=0$, then $0x^2+0x+0$ is not a part of $P_2$? Or is ...
-1
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0answers
18 views

find the matrix and the extremum 0f matrix ,location,definitess [on hold]

You are given the following quadratic function. $$ Q(x,y,z)=3x^2-6x+6xz+y^2-4yz+8z^2 $$ Find the matrix associated with the extremum (minimum, maximum or saddle point). Determine the definiteness ...
1
vote
1answer
15 views

Prove that if $p\le n$, then $p$ does not divide $n! + 1$

I'm having trouble on how to approach this problem Prove that if $p\le n$, then $p$ does not divide $n! + 1$ ($p$ is prime and $n$ is an integer).
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1answer
24 views

Is a vector of coprime ring elements column of a regular matrix ?

Given a commutative Ring $R$ with unit and $a_1=(r_1,\ldots,r_n)^T \in R^n$ with coprime entries (i.e. $\sum_i Rr_i=R)$. Are there $a_2,\ldots,a_n \in R^n$ such that the matrix $A = (a_1,\ldots,a_n) ...
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0answers
15 views

Transition Matrices for Jordan Form [duplicate]

Thought I would throw out my line one more time. I have this matrix $M$ $M = \begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & -1\\ 0 & -1 & 1 ...
0
votes
1answer
23 views

Operator and invertibility

Give an example of a vector space $V$ over $\mathbb R$, an operator $T \in L(V)$, and numbers $\alpha $, and $\beta $ such that $\alpha^2 < 4 \beta $ and $ T^2 + \alpha T + \beta I $ is not ...
2
votes
1answer
30 views

Every matrix in $SU(2)$ can be written as: $P= I\cos \theta+ A\sin \theta$, $A$ on the equator.

How can I show that every matrix in $SU(2)$ can be written as: $P=I\cos \theta + A\sin \theta$, with $A$ on the equator?
0
votes
1answer
16 views

Find projection of a function onto a subspace [on hold]

Consider the space $C[0,2\pi]$ of continuous functions on the interval $[0,2\pi]$ with the inner product $$(f,g)= \int_0^{2\pi} f(t)g(t)\ dt.$$ Find projection of the function $f(x)=2x$ onto the ...
1
vote
1answer
36 views

Unitary Matrices in Linear Algebra

Could anybody provide the examples of two unitary matrices which sum is also unitary Let A = $$ \left[ \matrix {1&0\\ 0&1\\}\right] $$ Then what would be B? I need to show that $ ...
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0answers
31 views

Diagonalization and $T(f(t))=f(t+1)$

Let $T \colon \mathbb{P}_n(\mathbb{R}) \to \mathbb{P}_n(\mathbb{R})$ defined as $T(f(t))=f(t+1)$. $T$ is diagonalizable? Why? I know that $1$ is eigenvalue of $T$. I did for case $n=2$. I do not know ...
0
votes
1answer
19 views

Systems of First Order Linear Equations, finding P(t) from two given vectors

Consider the vectors $x^{(1)}(t) = (t,1)$ and $x^{(2)}(t) = (t^2, 2t)$ I computed the Wronskian which is t^2. I also know that it's continuous everywhere except when t=0. But I was wondering how to ...
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votes
2answers
24 views

Find the projection p of x onto the span of u1 and u2

where $u_1=(2/3, 2/3, 1/3)$ and $u_2=(1/\sqrt2, -1/\sqrt2, 0)$ and $x=(1,2,2)$ how do I find the span of $u_1$ and $u_2$? after that do I just use the formula for the vector projection of x onto the ...
3
votes
2answers
18 views

Divergence of fixed-point iteration for real starting values

Consider the linear system of equations $Ax = b$ with invertible $A\in \mathrm{GL}(n,\mathbb R)$ and $b\in\mathbb R^n$. For $A = M - N$ with invertible $M$ the solution $x_* = A^{-1}b$ is a fixed ...
0
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0answers
13 views

Tomas Moller's Triangle-Triangle Intersection

I'm reading Tomas Moller's "A Fast Triangle-Triangle Intersection Test" (http://web.stanford.edu/class/cs277/resources/papers/Moller1997b.pdf) and am at a point where I'm not sure what he is talking ...
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0answers
6 views

Bounding L2 norm of a weighted matrix in terms of the L2 norm of the unweighted matrix.

Suppose $S=\sum_{i=1}^n x_ix_i^T$ be the covariance matrix, and suppose the $L_2$ norm is given $\|S\|=a$. Now let $w_1,\dots,w_n$ be a series of weights. Let $S_w$ be the weighted covariance matrix: ...
0
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0answers
27 views

Jordan Canonical Form transition matrix

I have this matrix $M$ $M = \begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & -1\\ 0 & -1 & 1 \end{bmatrix}$ And I was asked to put it into Jordan Canonical ...
0
votes
1answer
39 views

Find Eigenvalues and Eigenvectors of A

Let $\mathbf{A}\mathbf{x}=\mathbf{a} \times \mathbf{x}$, where $\mathbf{x} $ and $\mathbf{a}$ are in R$^3$ and $\mathbf{a}$ is a fixed or constant vector. Find the eigenvalues and eigenvectors of A.
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2answers
20 views

Transpose Operator is diagonalizable?

Let $T \colon \mathbb{M}_{nxn}(\mathbb{R}) \to \mathbb{M}_{nxn}(\mathbb{R})$ the linear operator such that $T(M)=M^t$, where $M^t$ is the transpose of the matrix $M$. Prove that $T$ is diagonalizable. ...
0
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0answers
27 views

How do I find vectors that are linear independent of another two vectors in $\Bbb R^5$

I am given two vectors in $\Bbb R^5$ $\vec x$, and $\vec y$ and told to find 2 vectors $\vec u, \vec v$ that are linear independent of $\vec x, \vec y$. $\vec x =(2,3,-7,4,1)$ $\vec y =(0,0,0,0,1)$ ...
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votes
2answers
24 views

Write the Jordan form of an operator

These are the properties that apply to the operator $A$. $k_A(x)=x^4(x-2)^4, d(A)=2, d(A^2)=4, d((A-2I))=2, (d((A-2I)^2)=3$ $d$ denotes the defect. $k_A$ is the characteristic polynomial. I ...
0
votes
1answer
13 views

Finding base of a subspace

Find base of a subspace and expand it to the base of $\mathbb{R}^4$ subspace is given by the following system of eqiuations: $ \begin{cases} x_1+2x_2+2x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$ ...
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votes
1answer
18 views

Disprove that this subset of P3 is not a subspace by using a counterexample

The set of all polynomials with degree 3 plus the zero polynomial. A hint would be appreciated to get me going :)
4
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0answers
29 views

Solving a recurrence with diagonalization?

Considering the recurrence $F_n=F_{n-1}+3F_{n-2}-3F_{n-3}$ where $F_0=0$, $F_1=1$ and $F_2=2$. Use diagonalization to find a closed form expression for $F_n$. So I first continued the recurrence to ...
0
votes
1answer
34 views

What textbook is being used in these lectures (Linear Algebra)?

I am learning Linear Algebra from these lectures by Prof. Adrian Banner (Princeton University) Does anyone know what textbook they are using? This is a link to the playlist on YouTube: ...
0
votes
1answer
21 views

How to show existence of an orthogonal map?

I want to show that the following holds: Let $x,y\in \mathbb{R}^n\setminus\{0\}$ be given and such that $\|x\|=\|y\|$. There is an orthogonal map $T$ such that $Ty=x$ (a rotation). How could one ...
0
votes
1answer
36 views

Optimization Problem (Linear Algebra)

I am not trying to cheat or anything, so any reference to online literature or MOOCs, that teach this stuff, will be highly appreciated. The problem is to prove that the following optimization ...