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 ...

learn more… | top users | synonyms

0
votes
0answers
7 views

Subspace of $ P_2?$

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

find the matrix and the extremum 0f matrix ,location,definitess

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
13 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).
0
votes
0answers
5 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) ...
-1
votes
0answers
9 views

Transition Matrices for Jordan Form

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
15 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
22 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? Also: How could I show that $\gamma_P(A)= A$ here?
0
votes
1answer
10 views

Find projection of a function onto a subspace

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
33 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 $ ...
1
vote
0answers
21 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
12 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 ...
-1
votes
2answers
16 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
17 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
votes
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 ...
0
votes
0answers
4 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
votes
0answers
19 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
38 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.
0
votes
2answers
17 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
votes
0answers
26 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 $\vec x, \vec y$. $\vec x =(2,3,-7,4,1)$ $\vec y =(0,0,0,0,1)$ ...
0
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
11 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}$ ...
0
votes
1answer
14 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 :)
3
votes
0answers
27 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
32 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
18 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
31 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 ...
0
votes
0answers
9 views

Show subspace can be rewritten as $n-k$ equations

Prove that every $k$ dimensional subspace $V \subset K^n$ can be described using $n-k$ linear equation. I think about applying Kronecker-Capelli theorem.
0
votes
0answers
17 views

When is “$\Re(\lambda) \gt 0$ for $\lambda \in \sigma(A),A \in \mathbb{R}^n $” true?

Let $A \in \mathbb{R}^{n \times n}$ and $\sigma(A)$ the spectrum of $A$. I am searching for a fast way to check whether $\Re(\lambda) \gt 0$ for all $\lambda \in A$. If $A = A^t$, one only has to ...
0
votes
0answers
17 views

Minimal polynomials and cyclic subspaces

I'm trying to make my way through two problems in Curtis's Linear Algebra, chapter 25. One of the two problems is this one, #5: Prove that $V$ is cyclic relative to a linear transformation $T \in ...
1
vote
0answers
12 views

If $M:=i\mathbf{\sigma}\cdot \mathbf{v}$, how do I see that $SMS^{-1}=i\mathbf{\sigma}\cdot \mathbf{v}'$

Let $\mathbf{\sigma}=\sigma_1+\sigma_2+\sigma_3$, where the $\sigma_i$ are the Pauli matrices and define: $$M:=i\mathbf{\sigma}\cdot \mathbf{v}$$ The claim is that if I change $M$ through a ...
0
votes
1answer
30 views

Algebra: Help with these expressions about inverse matrix

$x'$ meaning transpose of vector $x$. Let's say I have this expression: $$(x\cdot v')^2$$ I can write it as: $$ (x\cdot v')^2 = (x \cdot v')(x \cdot v') $$ My question came up when I saw somewhere ...
2
votes
1answer
32 views

Finding the Jordan basis of a linear map

A linear map $A$ is given in the canonical basis with the matrix $$ \begin{bmatrix} -2&0&-2&-2\\ 1&0&1&1\\ -1&1&-1&-1\\ 3&-1&3&3\\ \end{bmatrix} $$ ...
-1
votes
0answers
26 views

The set of matrices with nonnegative determinant is not a subspace. [on hold]

Disprove using a counterexample: The set of all $3\times 3$ matrices with determinant $\ge 0$ is a subspace of $M_3(\Bbb C)$.
0
votes
1answer
15 views

Let T:V->W be linear, show KerT is a subspace of V and imT=T(V) is a subspace of W

Ok so I have already proven that KerT is a subspace of V, which is pretty obvious because the kernel is just the 0's, though I'm not sure I did it formally enough. The second part I don't know how to ...
1
vote
1answer
26 views

show there exist non zero vector which is linear combination of other

sLet $a_1, \ldots , a_n$ be a basis of linear space $V$ let $W \le V$ be a $k$ dimensional subspace $k \ge 1$ Show for each subset $\displaystyle a_{i_i}, \ldots a_{i_m}$ for $m>n-k$ exist non ...
1
vote
1answer
21 views

Equation for minimum/maximum eigenvalue

It is well known that for a hermitian matrix $A$ we have $\lambda_{min}(A)=min{x\ne 0} <x,Ax>/<x,x>$, which we can see be diagonalizing $A$. Now here is my question about the following I ...
0
votes
3answers
26 views

“The limit of a sequence is insensitive to finite changes in the sequence” - help me understand this sentence!

The context is p59 of "A primer on Hilbert Space Theory, by Alabiso and Weiss, where the example is given of a quotient $c_{00}/c_0$ which is "the quotient of the space $c_0$, the space of all ...
0
votes
1answer
35 views

Prove that $tr(A^-)=\sum_{i=1}^n\lambda_i^{-1}$ [on hold]

If $A$ is an n$x$n symmetric matrix with $r$ nonzero characteristic roots $\lambda_1,\lambda_2,...,\lambda_n$ and $A^-$generalized inverse of $A$ (not $A^{-1}$), then ...
0
votes
3answers
40 views
1
vote
0answers
87 views

Typo in the book or am I going crazy?

I am reading about integral bases from Frazer Jarvis' "Algebraic Number Theory", but my question is really about elementary linear algebra. In page 49, author claims the following: I don't think ...
2
votes
0answers
19 views

What is wrong with this parametrization?

I need to find the $N$-by-$1$ vector $\mathbf{x}$ that minimizes the following expression: $L=\alpha |\hat{\mathbf{H}}_{1}\mathbf{x}|^2 +(1-\alpha)|\mathbf{H}_{2}\mathbf{x}-\hat{\mathbf{Y}}_{2}|^2$, ...
0
votes
0answers
28 views

Does Linearity imply Commutativity? [on hold]

If I have two linear operation X and Y , could I conclude that for X + Y = Y + X ?
-1
votes
0answers
16 views

How to prove the operator D=d^(4)/dx is self adjoint

I'm trying to prove $D=d^{4}/dx$ is self adjoint, I think it is trivial but the book let me use Lagrange identity to show it.
0
votes
1answer
12 views

Rational Canonical Form Confusion; Choosing Basis Which Gives the Rational Canonical Form.

I am reading the theory of finitely generated modules over a PID. One of the applications of the the theory is that one can derive the theory of rational canonical form of a linear operator on a ...
0
votes
2answers
63 views

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____?

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____? This is a fill-in-the-blank problem that I found in my paper, but I don't have this answer.
0
votes
0answers
14 views

Non-orthogonal basis

I have a set of complex vectors (maybe 10,000 vectors, each of which has maybe 200 elements). I know that each of the complex vectors is a linear combination of a small (maybe 10) collection of ...
0
votes
2answers
31 views

Orthogonal Matrix with a specific row

I have an assignment with the following question: Does an Orthogonal Matrix exist such that its first row consists of the following values: ($1$/$\sqrt{3}$, ...
0
votes
3answers
33 views

Help understanding a proof about vector spaces

The exercise goes like this: -Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$. -Find the dimension $[\mathbb{R}^3|W]$ This was a problem from my algebra exam, ...
0
votes
0answers
26 views

Find the eigenvalues and eigenvectors of T in V

Let $\mathbf{V}$ be the linear span of the functions 1, cos x, sin x. Let the operator T on V be given by the rule $T y(x)= y(x+\pi/4)$. Find the eigenvalues and eigenvectors of T in V. I'm not sure ...
0
votes
1answer
25 views

What is the purpose of Jordan Canonical Form?

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. What is ...