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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2
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52 views

What is the point of basis vectors?

Why do we even bother with basis vectors? Why don't we just notate an element $x$ of an $n$-dimensional vector space $V$ as an ordered set $(x_1,x_2,...,x_n)$ and go from there?
5
votes
4answers
343 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = ...
0
votes
2answers
37 views

For which value of k will the vector be a combination of two vectors?

For which value of $k$ will the vector $\begin{bmatrix}1\\-2\\k\end{bmatrix}$ in $\mathbb{R}^3$ is a linear combination of the vectors $w=\begin{bmatrix}2\\-1\\-5\end{bmatrix}$ and ...
0
votes
1answer
68 views

Linear Algebra, multiplication of Tensor by vector by vector.

I am deriving some equations and need to know the correct mathematical notation for opening up the brackets of an equation with the following variables: tensor $A \in$ ${\mathbb R}^{l \times l \times ...
0
votes
1answer
28 views

a question about linear algebra and matrix

Given a $n\times n$ matrix A,and the matrix's characteristic polynomial is $|\alpha I-A|=(\alpha-a_{1})^{r_{1}}(\alpha-a_{2})^{r_{2}}...(\alpha-a_{p})^{r_{p}}$,and $r_1+r_2+...r_p=n$. Then,as for any ...
1
vote
0answers
10 views

Representation of Affine Maps

I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$: $$ f[\lambda x+ (1-\lambda) y]=\lambda ...
0
votes
0answers
5 views

Find all vertices of a parallelepiped given only 3 to start with (linear algebra)

My question is simple. I just want to find out the rest of the vertices given only three of them. I haven't really grasped the process of finding the vertices, so I need someone to help me understand ...
0
votes
0answers
10 views

Find an orthogonal basis of inner product

Let's define dot procduct $<A,B>=Trace(A B^T)$ over $M_{n \times n}(\mathbb{R})$ Find basis or system of equations describing an orthogonal $W^\perp$ subspace to subspace $W$ which consist of ...
0
votes
2answers
454 views

Change of basis matrix for orthogonal bases

I am trying to show that if $B_1$ and $B_2$ are orthonormal bases for $\mathbb{R}^n$, then the change of basis matrix $P$ from $B_1$ to $B_2$ is an orthogonal matrix. I'm a bit stuck. I started with ...
-1
votes
0answers
19 views

what is number of invertible matrix m*m on$ Z_n$?

‎‎please help me what is number of invertible matrix $‎m*m$‎‎‎ on Group $\mathbb{Z}_n$ ?‎‎‎, assuming we know‎ this number in $\mathbb{Z}_p \quad$ is $‎(p^{n}-1)(p^{n}-p) \cdots ...
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2answers
47 views

Propositions of elementary matrix

i'm trying to solve a question about elementary matrix. When given $A_{m,n}$ and $B_{n,p}$ which differ from the Zero matrix. Also, multiplying of $A$ and $B$ is the zero matrix, that is: $AB=0$; ...
0
votes
1answer
13 views

Simple modules over R isomorphic to R/I

Let $R$ be a ring, and let $M$ be a simple $R$-Module, meaning that it only has the trivial submodules {0} and $M$. Show that there's a maximal ideal $I \subset R$ so that $M \cong R/I$. Thanks in ...
-6
votes
2answers
29 views

simultaneous equation question [on hold]

2x - 3y = 6 || -⅔x + y = 1 what is x and y ?
0
votes
1answer
29 views

motivation and theoritical clarification of some linear transformation related concepts

I am self studying linear transformation but i am surprised to be introduced with the ideas of KerT and ImT just after the introduction of the definitions and examples of linear transformation.So my ...
2
votes
1answer
483 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
0
votes
1answer
25 views

(direct) sum vs span of subspaces

Is there a difference between the span of subspaces and the sum of them in linear algebra? They both seem to just be the set of all linear combinations.
2
votes
3answers
57 views

Find the inverse of a submatrix of a given matrix

I have a $A$ matrix $4 \times 4$. By delete the first row and first column of $A$, we have a matrix $B$ with sizes $3 \times 3$. Assume that I have the result of invertible A that denote $A^{-1}$ ...
1
vote
2answers
27 views

Eigen space of $T_{A}$ where $T_{A}(v)=Av$

Let $T_{A}:\mathbb{C}^3\rightarrow \mathbb{C}^3$ ,$T_{A}(x,y,z)=A\begin{pmatrix}x\\ y\\ z\end{pmatrix}$ and $A=\begin{pmatrix} 1 & 5 & 0\\ 0 & 1 & 0\\ 0&0&3 \end{pmatrix}$. I ...
0
votes
1answer
10 views

Is there any shortcuts in getting an H-infinity norm of a matrix expression?

One of the past exam problems I was solving, has this in its official solution: Usually, to calculate the $H_{\infty}$ norm of any matrix expression $M$ I'd first calculate the eigenvalues of ...
-1
votes
1answer
11 views

properties of orthonormal systems and hilbert spaces [on hold]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
0
votes
0answers
28 views

What is my error in this matrix / least squares derivation?

I'm doing a simple problem in linear algebra. It is clear that I have done something wrong, but I honestly can't see what it is. let, $y = Ax$, $y_{ls} = Ax_{ls}$ where A is skinny and full rank, ...
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votes
0answers
44 views

Which of these are true? And why? [on hold]

Which of these are true? And why are they true, please answer this too, if possible?
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0answers
9 views

How to decrypt a ciphertext by using the mutual index of coincidence?

I am trying to decrypt a Vigenére cipher text. I have found the key length by computing Index of Coincidence of substrings. The key length is 12. I know the letter frequencies the string and the ...
-1
votes
0answers
10 views

Show that W is a Subspace of R³

can you help me: Let $u=(1,2,-3)$ and $v=(-2,3,0)$ Two Vectors in R³ and let W the subpace of R³ that consists of all the vectors shape $au+bv$, where, $a,b ∈ R $ show that W is subspace of R³ im ...
3
votes
2answers
51 views

Eigen value and transpose of the $Matrix$

Let $A$ and $B$ are linear operator on a finite dimensional vector space $V$ over $\mathbb R$ such that $(AB)^2 = AB$ and $BA$ is invertible , then which of the following are true ? $AB = ...
5
votes
1answer
35 views

Write $\mathbb{P}^3_{\mathbb{C}}$ as a union of disjoint lines

Is there a set $\Gamma=\{L \subseteq \mathbb{P}^3_{\mathbb{C}}: L \textrm{ is a projective line}\}$ such that every point $p \in \mathbb{P}^3_{\mathbb{C}}$ lies on exactly one line $L_p \in \Gamma$? ...
1
vote
1answer
26 views

Connection between algebraic multiplicity and dimension of generalized eigenspace

Assume $V$ to be a finite dimensional vector space. Define the algebraic multiplicity $am(\lambda)$of an eigenvalue $\lambda$ of a linear operator $T:V\to V$ as the maximum index of the factor ...
0
votes
1answer
39 views

Eigenvalue of $B=uv^\text{T}+wz^\text{T}$

We have $u,v,w,z \in R^\text{n}$, how can we express the eigenvectors and eigenvalues of $B=uv^\text{T}+wz^\text{T}$ by analyzing over $u,v,w$ and $z$?
12
votes
2answers
1k views

If $A^2$ and $B^2$ are similar matrices, do $A$ and $B$ have to be similar?

I know that the converse is true; that is, if A and B are similar matrices, then $A^2$ and $B^2$ are similar . However, I'm not sure about the reverse.
0
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0answers
19 views

Annihilating polynomials of a linear operator

Does the annihilating polynomial of a linear operator reduce it to the zero transformation ? What are the applications of such polynomials ?
0
votes
1answer
45 views

Prove there exists a unique map T

Let $V_i$ be a collection of vector spaces over field $F$ where $i=1,2,...,N$. Given the Cartesian Product $V=V_1\times V_2\times...\times V_N$ equipped with natural projections $p_i:V\to V_i$. ...
3
votes
2answers
22 views

Getting perpendicular distance by Gram Schmidt Process

Use the Gram-Schmidt process to find the perpendicular distance from the points to the corresponding lines in the problems. a. point $(0,0)$ to the line through $(1,1)$ and $(3,0)$ b. point $(-1,0)$ ...
3
votes
2answers
39 views

Prove that matrix is symmetric and positive definite given the fact that $A+iB$ is.

I have some questions regarding the following problem Let $ A + iB $ - hermitian and positive definite, where $A, B \in \mathbb R^{n\ \times\ n} $ show that the real matrix $$C =\begin{pmatrix} A ...
2
votes
0answers
24 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
0
votes
2answers
39 views

Jacobian of n linearly independent forms in n variables

Let $k$ be a field of characteristic zero and let $f_1, \ldots, f_n \in k[x_1, \ldots, x_n]_d$ be linearly independent forms of degree $d$ in $n$ variables. Is there a nice algebraic argument for ...
0
votes
0answers
21 views

covering finite Dimensional vector space

can a finite dimensional symplectic vector space over finite field be covered with mutually transversal Lagrangian planes(maximal Isotropical Subspaces )?
9
votes
3answers
313 views

Calculate a determinant.

Let $a_{1}, \cdots, a_{n}$ and $b$ be real numbers. I like to know the determinant of the matrix $$\det\begin{pmatrix} a_{1}+b & b & \cdots & b \\ b & a_{2}+b & \cdots & b ...
0
votes
1answer
43 views

Abstract Linear Algebra Inner Product [on hold]

Let $u\in\mathbb{R}^n$ be a vector such that $\|u\|=1$ (for the usual inner product). Prove that there exists an $n\times n$ orthogonal matrix whose first row is $u$.
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0answers
14 views

How to compute the best fitting frustum for a set of points?

I am struggling with a problem that I am sure is well known, but I could not find any answer using google or searching on MathOverflow. I have a set of 3D points (x,y,z) and a camera reference frame ...
1
vote
1answer
455 views

How do you diagonalize this matrix and find P and D such that A = PDP^-1?

1 1 4 0 -4 0 -5 -1 -8 I3 = 3x3 identity matrix λ 0 0 λI3 = 0 λ 0 0 0 λ λ-1 -1 -4 = 0 λ+4 0 5 1 ...
0
votes
2answers
28 views

How to show if 2 vectors are the same

I've been given 2 lines in different forms $L1$ is $$\frac{x-1}{4} = \frac{y-2}{3} = \frac{z-10}{5} $$ $L2$ is $$x = -7-4t$$ $$y = -4-3t$$ $$z = -5t$$ I've converted $L2$ into its Cartesian form as ...
0
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0answers
18 views

orthogonality of two distinct vectors [on hold]

Let L be the line containing the distinct vectors p and q and let r be a vector not on L such that (u-r) is perpendicular to (q-p) is a solution of the simultaneous equations: (u-r, q-p) = 0 u = ...
3
votes
4answers
81 views

Polynomial: Is there a theorem that can save my proof when $K$ doesn't include $\mathbb C$

Suppose $f(x),g(x)\in K[x]$ ($K$ a number field), let $f(x)=x^{3m}+x^{3n+1}+x^{3p+2}$, where $m,n,p\in\mathbb N$, and let $g(x)=x^2+x+1$, prove: $$g(x)\mid f(x)$$ I think this problem is not ...
1
vote
1answer
40 views

$A$ is the set of all $n \times n$ matrices where $\operatorname{tr}(A)=0$, is $A$ a subspace of $M_{nn}$ (where $n\ge2$)?

$\newcommand{\tr}{\operatorname{tr}}$For $A =$ zero matrix, $$W=\{ A \in M_{nn} : \tr(A) = 0 \}$$ I can proof that the set of all n x n matrices A with $\tr(A)=0$ is a subset of $M_{nn} $for$ \ n \geq ...
0
votes
1answer
26 views

Finding the matrix of a projection map and finding the eigenvalues

We have that $P$ is the orthogonal projection onto the span of $(v_{1},v_{2})$ with $v_1 = (\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2})$ and $v_2 = ...
1
vote
2answers
61 views

Largest eigenvalues of AA' equals to A'A

I need help with proving that for any real matrix,the largest eigenvalue of AA' equals to the largest eigenvalue of A'A
0
votes
1answer
21 views

Linear Form on a n-dimensional vector space $V$

Question : Show that if $f_1 , f_2, \ldots ,f_m $ are linear forms on a n-dimmensional vector space $V$ over $K$, then $\dim \ \cap_{i=1}^m \ker(f_i) \geq n-m$ with equality iff $f_1 , f_2, \ldots ...
0
votes
1answer
17 views

Let $P$ denote a subset of a linear space $L$, why does the set $P$ always contains a basis of $span\,P$.

Let $P$ denote a subset of a linear space $L$, why does the set $P$ always contains a basis of $span\,P$. In my opinion it is not the case. For example if $P$ consists of two vectors $(1, 0)$ and ...
2
votes
0answers
81 views

An Inequality in Linear Algebra (Stuck in the Proof)

The following question was on an exam which I didn't manage to solve. Later, I saw that the same question appears in page 146 of Kostrikin and Manin book: Suppose that $f:V\to V$ is an operator in a ...
3
votes
1answer
137 views

Proof for real Jordan canonical form

Let A $\epsilon$ Mat(nxn, $\mathbb R$) be a matrix that is diagonalizable in $\mathbb C$ with k real eigenvalues of algebraic multiplicity 1 and (n-k)/2 pairs of complex-conjugated eigenvalues of ...