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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Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
2
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1answer
49 views

Show that a Hilbert space with two inner products has a basis that is orthogonal with respect to both inner products

Let $\mathcal{H}$ be a complex, $n$-dimensional Hilbert space with two inner products $\langle \cdot, \cdot \rangle_1$, $\langle \cdot, \cdot \rangle_2$. Show that there exists a basis $ X = x_1, ...
10
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3answers
116 views

Is there such a thing as “quadratic independence” (and higher generalizations of linear independence)?

The notion of linear independence is very well-known and well-understood. However, is there a way to generalize the definition to other types of independence -- such as perhaps "quadratic ...
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1answer
60 views

Calculating Eigenvalues is only

Assume that the following is used: $$ A = \begin{pmatrix} 0& 1&\\ 2& 3&\\ 4& 5&\\ 6& 7&\\ 8& 9& \end{pmatrix} $$ Then calculating the Coveriance ...
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0answers
55 views

Prove that every proper principal submatrix of $\lambda I-A$ is nonsingular under certain assumptions

Given that $A$ is a complex square matrix of order $n$, $\lambda$ is an eigenvalue of $A$ with geometric and algebraic multiplicity $1$, and $x,y$ are entrywise nonzero vectors such that $Ax=\lambda ...
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1answer
24 views

Matrix addition and definiteness

Is strict/weak negative/positive definiteness/semidefiniteness of matrices preserved under matrix addition? I tried to do this for 2x2 matrix but even this wasn't easy. (I tried to use the principal ...
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2answers
41 views

Interpretation of Powers of matrix

Suppose there is a square binary matrix (Adjacency matrix of a graph), $A$. I got that, the matrices, $A^2$ and $A^3$ are distinct but the set of eigenvalues are same for $A^2$ and $A^3$. It is to be ...
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3answers
289 views

Understanding underlying algebra behind simplified expression

The solution to a linear algebra problem I'm working on reads: $$\det(A-\lambda I) = \det\begin{pmatrix}-\lambda & 1 & 0 \\ 0 & -\lambda & 1\\ 1 & -1 & ...
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1answer
40 views

Question on matrix

Suppose I have a vector field $F(x)=Ax$ where $A$ is a matrix. How can I express $Sx$ without $A$ (use $F$ instead)? Here $S=\dfrac{A+A^T}2$ is symmetric part of $A$.
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1answer
321 views

Minimal polynomials and degree of field extension

I have a cyclotomic field $\mathbb{Q}(\zeta_3)$, and want to know how I can find a minimal polynomial of $\zeta_{10}$, and $\zeta_{12}$. I have determined that both the polynomials should be of ...
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1answer
234 views

inverse of an infinite matrix

How to find inverse of an infinite lower triangular matrix all of whose diagonal entries are 1 and the entries of each column are given by coefficients of some power series rings?
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1answer
35 views

How to write this function?

I do not want the answer given to me, I just want assistance. Problem: Marcus invests $750 in an account that pays 9.8% interest compounded annually. Write a function that describes the account ...
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votes
1answer
65 views

How does LU decomposition work?

I'm interested in the algorithm of LU decomposition in order to solve a LSE like $Ax=b$, where $A$ is a square matrix. My question is: When I compute $PA=LU$ do I also need to interchange rows in $L$ ...
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0answers
39 views

Is this triangle construction possible?

I must construct this triangle: Consider the triangle $ABC$. Take $D$ in the line of $BC$ such that $C$ is the mid point of $BD$ and take $Y$ in the line $AC$ such that the lines $AB$ and $BY$ are ...
5
votes
1answer
97 views

Characterisation of normal matrices

How to prove the following? Lemma. Let $C=[A,A^{\star}]$. $A$ is normal iff $[A,C]=0$. One direction is trivial. The other direction reduces to showing that $A^2 A^\star+A^\star A^2=0$ implies that ...
2
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1answer
34 views

Vectors linearly independent implies sum of vectors also L.I.

I have that the set of vectors: $$\vec u, \vec v, \vec w$$ is L.I. This means that: $$a_1\vec u + a_2\vec v + a_3\vec w = \vec 0 \implies a_1 = a_2 = a_3 = 0$$ I need to prove that the set: ...
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4answers
102 views

Equation of a line that passes halfway between two points (in other words, divides the space)

Is there a formal proper way of finding the line between two points? By that I don't mean the line connecting the two points, I mean a line that runs the same distance away from point 1 and point 2. ...
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votes
1answer
75 views

Additive inverse

Hey so I got a question about Vector spaces Let $V=(8,\infty)$. For $u,v$ in $V$ and $a$ in $\mathbb R$ define vector addition by $u\boxplus v:= uv-8(u+v)+72$ and scalar multiplication by ...
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2answers
43 views

Find a Subspace of $\mathbb{R}^3$ with dimension $m$.

For each $m \in \{0,1,2,3\} $ find a subspace of {0,1,2,3} of dimension $m$ and verify answers. I'm not sure what is meant by this or how to begin solving?
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1answer
32 views

differentiation linear map and matrices

let $P_5 = \{ a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 \}$ be the vector space of polynomials of degree $\leq 5$ over $\mathbb{Q}$. Denote $D: P_5 \to P_5$ as the differentiation linear map, ...
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2answers
38 views

A question from hoffman y kunze. About Projections

Find a projection $E$ wich projects $\mathbb{R}^2$ onto the subspace spanned by $(1,-1)$ along the subspace spanned by $(1,2)$. What is the way to approach this problem? Almost to start! Any ...
2
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1answer
34 views

Critical eigenvalues for smooth family of 2x2 matrices?

Consider the following simple setup: we have a smooth family of symmetric $2\times 2$ matrices $A(t)$, with normalized eigenpairs $(\lambda_1(t),v_1(t))$ and $(\lambda_2(t),v_2(t))$. Suppose there ...
2
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2answers
72 views

Show $f(W) \subset W \Rightarrow f(W^\perp) \subset W^\perp$ and $f(\langle v \rangle ) \subset \langle v \rangle \Rightarrow$ eigenvector [closed]

Let $f \in \textrm{O}(n, \mathbb R)$. (O is the orthogonal group) Show: i) If $f(W) \subset W$ for a subspace $W \subset \mathbb R^n$, then $f(W^\perp) \subset W^\perp$. ii) If $f(\langle v \rangle ...
7
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0answers
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When is every matrix in the span of two matrices singular? [duplicate]

Given two square matrices $A, B$, when is $$\det(A+tB) = 0$$ for all $t\in \mathbb{R}$? An easy sufficient condition is that $A$ and $B$'s kernels have nontrivial intersection. Per Henning's comment ...
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votes
1answer
115 views

Multiply a circulant matrix by a vector with FFT.

I am asked to write a Matlab program to find the coefficients of the resulting polynomial which is the product of two other polynomials. However, I need someone to clarify the underlying concepts for ...
4
votes
1answer
63 views

Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
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votes
3answers
88 views

Solve the following system of linear equations for any values of real parameter $a$…

For any values of parameter $a$ solve the following system of linear equations: $$\begin{cases} x+y+2z=1 \\ 2x+ay-z=4 \\ 3x+y+3z=1 \end{cases} $$ Calculating the value of determinant I found out, ...
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1answer
15 views

Consider the action of $S_3$ on $C^3 = \{ (x,y,z) | x + y + z = 0\}$. Show that $\rho$ is irreducible.

The action is defined as $\rho_g (x_1, x_2, x_3) = (x_{g(1)}, x_{g(2)}, x_{g(3)})$. For example: if $g=(12)$, then $g(2,3,-5) = (3,2,-5)$. I understand that the action just permutes the elements, ...
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1answer
32 views

Relationship between vectors of a triangle and dot product

So I have to make this exercise: $\vec u + \vec v + \vec w = 0$ I also know that $||\vec u|| = \frac{3}{2}, ||\vec v|| = \frac{1}{2}, ||\vec w||=2$. Now take: $$\vec u \cdot \vec v + \vec v \cdot ...
2
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0answers
126 views

Sylvester Determinant and Resultant (Homework)

I have a homework problem where I have to show that if $f$ and $g$ are polynomials with no common factor in $K[x, y]$, where $K$ is a field, there are only finitely many elements $(a, b)\in K^2$ such ...
2
votes
1answer
31 views

$S$ as a subspace over $\mathbb R$ vs. as a subspace over $\mathbb C$?

Let $S \subset \mathbb C^n$ be an over-$\mathbb R$-defined subspace of $\mathbb C^n$ and $B=\{b_1, \ldots, b_m \}$ an $\mathbb R$-basis of $S_{\mathbb R}$. Show that $B$ is also a $\mathbb C$-basis ...
2
votes
2answers
230 views

Normal but not hermitian nor unitary

I have to find out a normal transformation that is neither hermitian nor unitary. http://en.wikipedia.org/wiki/Normal_matrix gives me the answer. However, I would like to know how to find it out ...
2
votes
1answer
78 views

Identify nilpotent matrix according to its characteristic polynomial (all eigenvalues are $0$)

I was wondering about something. Say $A_{n \times n}$ is a matrix and it's characteristic polynomial is $P(x)=x^n$ (all eigenvalues are $0$), can you say that $A$ is a nilpotent matrix? I really ...
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1answer
34 views

$A\in GL_n(\mathbb Z)$ is a product of elementary matrices $E_1, \ldots, E_k \in GL_n(\mathbb Z)$

I'd like to show that if a matrix $A\in \mathbb Z^{n\times n}$ is invertible (with $A^{-1}\in \mathbb Z^{n\times n}$) then $A$ can be written as a product of elementary matrices $E_1, \ldots, E_k \in ...
1
vote
1answer
75 views

Is the empty set is a subspace of any vector space

Is the empty set is a subspace of any vector space? im not too sure about this one, is the zero vector in the empty set?
2
votes
2answers
75 views

A non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root

Is there any quick argument to show that every non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root? Just the existence without computing it. Knowing that $A\in \mathbb C^{2 \times2}$ ...
3
votes
1answer
164 views

Eigenvalues of product of symmetric positive-definite matrix

Let $M$ be a symmetric positive-definite matrix and $$A = (I+M)^{-1}(I-M)$$ we know that eigenvalues of matrices $I+M$ and $I-M$ are as $1+\mu_i$ where $\mu_i$ is eigenvalue of $M$. Who we can ...
3
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0answers
54 views

How to prove this identity?

Let $$ x_1=- \frac{b\, t}{\left(a + e\, t\right)}, \\ x_2=- \frac{c\, s\, \left(a + e\, t\right)}{\left(a\, b + a\, f\, s - b\, h\, s\, t + e\, f\, s\, t\right)}, \\ x_3=- \frac{q\, \left(a\, b + a\, ...
0
votes
2answers
1k views

Linear Algebra and planes in Cartesian space

I was asked this question from the course Linear Algebra and I need to show all working. The question is in 5 parts: Consider the xyz-space R3 with the origin O. Let l be the line given by the ...
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vote
1answer
258 views

Find an ordered basis of $V$ such that $[T]_\beta$ is a diagonal matrix.

The entire problem statement is: Let $V$ be a finite dimensional vector space and $T:V\to V$ be the projection of $W$ along $W'$, where $W$ and $W'$ are subspaces of $V$. Find an ordered basis ...
3
votes
4answers
102 views

Show if $A$ has a zero row, then $AB$ has a zero row.

Let $A$ and $B$ be $n \times n$ matrices. Show that if the $i$th row of $A$ has all zero entries, then the $i$th row of $AB$ will have all zero entries. Also give and example using $2 \times 2$ ...
2
votes
2answers
55 views

finding the value of u of equation 5u^2 = 10u

I was solving a question, and while solving that problem I noticed something $5u^2 = 10u$ (solving this) this can be solved as: $5 \cdot u \cdot u = 10 \cdot u$ $u = \dfrac{10u}{5u}$ $u = 2$ ...
7
votes
1answer
481 views

Why is Householder computationally more stable than modified Gram-Schmidt?

I'm having trouble nailing down why using Householder transformations yields a more stable/accurate result than the modified Gram-Schmidt method when computing the QR decomposition of a matrix. Can ...
2
votes
3answers
111 views

maximum value of $\det(A)$, elements $0, 1, 2, 3$,

$A$ is a $3\times 3$ real matrix, whose elements can be $0, 1, 2, 3$. What is the maximum value of $\det(A)$? $\det(3I)=27$, the maximum value should be $\gt27$. Thank you very much for your ...
0
votes
2answers
29 views

Let $V$ be Vector Space $C [-4,7]$ and $S$ consists of functions of the form $ae^{bx}$, and ($a,b$ are real constants). Is $S$ a subspace?

($a,b$ are real constant). it seems to me that it doesn't satisfies closed under addition. let $q=ce^{dx}$, $p=se^{tx}$, I couldn't transform $q+p$ into the form of $ae^{bx}$ and also, I don't know ...
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1answer
44 views

Nullspace of linear transformation

I've gotten a little help on the following problem, but I'm still having trouble with it: Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation. We define the nullspace of ...
4
votes
1answer
180 views

Show that $A=0 \iff \mathrm{tr}(A)=0$ where $A= M_1+ \cdots +M_{\ell}$.

Let $G=\{M_1, M_2, \ldots ,M_{\ell}\} \subset \mathcal{M}_n(\mathbb{R})$, such that G form a group for the usual matrix multiplication. Denote $A= M_1+ \cdots +M_{\ell}$. Show that $$A=0 \iff ...
2
votes
1answer
47 views

Condition for convergence

Let $A \in \mathbb{R^{m\times{n}}}$ with full row rank. Let $B=I-\lambda A^T(AA^T)^{-1}A$ with $\lambda \in \mathbb{R}$. Determine the set of values of $\lambda$ for which $\exists \lim_{k \to ...
1
vote
1answer
58 views

Proving $\displaystyle rang(AB) \le \inf(rang(A),rang(B))$

Supposing $\displaystyle A\in \mathbb{M}_{np}(\mathbb{R})$ and $B\in\mathbb{M}_{pq}(\mathbb{R})$: How can prove that: $\displaystyle rang(AB) \le \inf(rang(A),rang(B))$
2
votes
3answers
73 views

Finding the eigenvalues and eigenvectors of $4\times 4$ matrix

Find all eigenvalues and eigenvectors(and generalized eigenvectors) of the following matrix. $$\mathbf{A} = \begin{pmatrix} -1&0&0&0\\ 5&-2&0&0\\ 0&3&1&0\\ ...