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

2
votes
1answer
30 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
42 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 ...
0
votes
1answer
19 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]}$ ...
0
votes
1answer
32 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 ...
1
vote
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 ...
1
vote
0answers
32 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 ...
8
votes
3answers
25k views

shortcut for finding a inverse of matrix

I need tricks or shortcuts to find the inverse of $2 \times 2$ and $3 \times 3$ matrices. I have to take a time-based exam, in which I have to find the inverse of square matrices.
0
votes
1answer
19 views

Computing an orthonormal basis

In $R^3$, find an orthonormal basis for the subspace $[span ((1,1,1))]^\perp$. I just want to make sure that my answer is right. Let $V$ be a subspace of $R^3$ spanned by $(1,1,1)^\perp$. Then, ...
0
votes
1answer
25 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 ...
0
votes
4answers
53 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 ...
1
vote
0answers
13 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. ...
1
vote
1answer
22 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?
0
votes
1answer
37 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 ...
0
votes
1answer
34 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 ...
0
votes
3answers
25 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 ...
1
vote
0answers
11 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 ...
1
vote
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 ...
2
votes
1answer
31 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
24 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 ...
-1
votes
0answers
31 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
17 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).
-1
votes
0answers
16 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 ...
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 $ ...
0
votes
1answer
38 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 ...
2
votes
1answer
242 views

Show that $T-iI$ is invertible when $T$ is self-adjoint

Let $T$ be a self adjoint operator on a finite dimensional inner product space $V$. Then $ \| T(x)\pm ix \|^2=\| T(x) \|^2+\| x\|^2$ for all $x \in V$. Deduce that $T-iI$ is inverible. ...
0
votes
0answers
16 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 ...
1
vote
1answer
51 views

How to put a matrix in Jordan canonical form, when it has a multiple eigenvalue?

I have a question that reads: Put the matrix \begin{bmatrix} 3 & -4\\ 1 & -1 \end{bmatrix} in Jordan Canonical Form. Moreover, in each case, find the appropriate ...
1
vote
0answers
13 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
1answer
41 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
1answer
41 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
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
votes
0answers
29 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 $\vec x, \vec y$. $\vec x =(2,3,-7,4,1)$ $\vec y =(0,0,0,0,1)$ ...
-1
votes
0answers
37 views

Rational Canonical Form Question [on hold]

If I am looking for the Rational Canonical Form over the Real field for a matrix but have complex eigenvalues, how would I go about doing so? Any help is appreciated, thanks! Also, its for a 3 x 3 ...
0
votes
2answers
26 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 ...
5
votes
0answers
174 views
+250

How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
4
votes
0answers
51 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
15 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
27 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 :)
0
votes
1answer
49 views

Adjoint Operator and Inverse

I am solving the following question and I am not really sure about the way I approach Question 1: Assume that $T:U\rightarrow U$ is invertible map. Prove that $(T^*)^{-1}=(T^{-1})^*$ Here is my ...
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

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
0answers
11 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.
1
vote
0answers
25 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
1answer
14 views

column vector dot product with transpose

Say I have an orthonormal base, $B = \{v_1, v_2, \ldots v_n\}$ for space $\mathbb R^n$. Assuming $v_1$ is a row vector, what is $v_i^t \cdot v_i$ ? Is it a scalar, or is it an $n\times n$ matrix ...
2
votes
1answer
33 views

Finding the Jordan basis of a linear map

A linear map $A$ is given in the canonical basis with the matrix $$ \begin{bmatrix} -2&0&-2&-2\\ 1&0&1&1\\ -1&1&-1&-1\\ 3&-1&3&3\\ \end{bmatrix} $$ ...
0
votes
1answer
35 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 ...
1
vote
1answer
28 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 ...
3
votes
1answer
46 views

Formula for Determinant of Vectors given in spherical coordinates

In 2D, one has an easy formula for the determinant of two vectors given in spherical coordinates, i.e. $\begin{vmatrix} \cos(\phi_1) &\cos(\phi_2)\\ \sin(\phi_1) &\sin(\phi_2)\end{vmatrix} ...
-1
votes
0answers
33 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
19 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 ...