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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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
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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
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2answers
229 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 ...
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1answer
70 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?
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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
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1answer
159 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\, ...
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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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1answer
257 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 ...
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votes
4answers
101 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 $2 \times 2$ ...
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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
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1answer
480 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
110 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 ...
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2answers
27 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 ...
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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 ...
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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 ...
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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: $\displaystyle rang(AB) \le \inf(rang(A),rang(B))$
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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\\ ...
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1answer
182 views

Question about maximal orthonormal subset in infinite dimensional vector space

The question is this: Let $A$ be an orthonormal subset of vectors in an infinite dimensional vector space $V$. Suppose for every $0\neq y \in V-A$ there is a $v \in A$ so that $\langle v,y ...
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1answer
50 views

The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$

Let $X$ be irreducible algebraic set of projective n space. I am trying to show that: The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$, where $G(k,n)$ is the ...
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1answer
49 views

$T : \mathbb{R}^{5} \rightarrow \mathbb{R}^{2}$ with: $\mathrm{span}[v_{1}, v_{2}, v_{3}] = \mathrm{null}(T)$

I'm supposed to find the linear transformation given these 3 vectors. EDIT: they are linearly independent! sry I have just recently learned about span, linear combination, linear independence and ...
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2answers
34 views

How do I rearrange this equation?

I'm solving an op-amp question and have simplified the variables for convenience : Rearrange $$\frac{a+b}{c} = \frac{b-d}{e}$$ To get $$b = \frac{ae + cd}{c + e}.$$ Can anyone help me out ?
5
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1answer
136 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such ...
7
votes
2answers
322 views

Circulant determinants

Suppose that $a_1,a_2,\ldots,a_n$ are $n$ distinct real numbers; is the following statement true? There is a permutation of $a_1,a_2,\ldots,a_n$, namely $b_1,b_2,\ldots,b_n$, such that the ...
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0answers
49 views

For which values is the quadratic a perfect square?

$f(x)=ax^2+bx+c$ is a quadratic polynomial, $a,b,c$ are natural numbers. For which values of $a,b,c$, $f(x)$ can be a perfect square? If it can be perfect square, for which integer values of $x$ it is ...
2
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2answers
67 views

Find $\xi$ such that matrix is unitary and find $\lambda_i$ such that matrix hermitian

i) Find $\xi \in \mathbb C$ such that $$\frac12 \begin{pmatrix} \xi & 0 & 1+i \\ 0 & 2 & 0 \\ 1-i & 0 & -1+i \end{pmatrix}$$ is unitary. ii) Let $A := ...
3
votes
1answer
151 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
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0answers
50 views

diagonal action induces permutation

Suppose one has two $n$-tuples of complex numbers $(c_1,\dots,c_n)$ and $(z_1,\dots,z_n)$ such that all $c_i$, $z_i$ are nonzero, and $$ (c_1z_1,\dots,c_nz_n)=(z_{\sigma(1)},\dots,z_{\sigma(n)}) $$ ...
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1answer
37 views

Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?

I wonder about the second step of the proof shown below (d) in the picture attached. Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?
2
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1answer
87 views

Is the wedge product surjective?

Is the wedge product $\wedge : \Lambda^{p}(V) \otimes \Lambda^{q}(V) \to \Lambda^{p+q}(V)$ surjective, for $V$ a real vector space of finite dimension? What dimension does its kernel have?
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1answer
33 views

Linearly independent subset of a spanning set

Given $V_1 + V_2 \in \operatorname{Sp} \{V_1,..,V_n\}$ and $V_1 \notin \operatorname{Sp}\{V_2,...,V_n\}$, prove that $\{V_2,...,V_n\}$ is linearly independent. Well, I know that $ V_1 + V_2 \in ...
2
votes
0answers
75 views

Law of large numbers for linear (quadratic) combinations of i.i.d. random variables

Let $(X_i)_{i\in\mathbb{N}}$ be i.i.d. real random variables with zero mean. By the law of large numbers $$\frac{1}{n}\sum_{i=1}^nX_i \to 0 \quad\text{(almost surely, in probabability...) as ...
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votes
1answer
20 views

Solutions for simultaneous equations

I need to solve the following system of equations: $ac-5bd=5$ $ad+bc=0$ So far I've managed to find (I'll put it as in {$a,b,c,d$}): {$0,-1,0,1$} {$0,1,0,-1$} {$5,0,1,0$}
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1answer
73 views

Removing multiples of 3 from number sequence

Say I had a basic odd number sequence: Where N is the number position in the sequence x = n*2-1; input 1,2,3,4,5,6 1,3,5,7,9,11 output How would I go about ...
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1answer
32 views

Does this span contain this vector?

Gilt $\begin{pmatrix}2\\-1\end{pmatrix}\in L\left(\begin{pmatrix}0\\0\end{pmatrix},\begin{pmatrix}1\\2\end{pmatrix},\begin{pmatrix}-2\\-4\end{pmatrix}\right)$? Does the span L contain the vector ...
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votes
2answers
118 views

Is every element of a complex semisimple Lie algebra a commutator?

Let $L$ be a (finite-dimensional) complex semisimple Lie algebra. Then we know that $L = [L,L]$. Is it true that every element of $L$ must be a commutator? Since a complex semisimple Lie algebra is ...
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1answer
87 views

Polynomial: Finding its value

If $a-b=3$, $a+b+x=2$, then find the value of $(a-b)\left(x^3-2ax^2+a^2x-(a+b)b^2\right)$ I could only substitute the value of $a-b$ there. I seriously want to try as much as I can on my own but ...
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1answer
82 views

Question about Bilinear form and inner product space

This is a question I have stumbled upon in a test I found on the web, and I don't even know how to approach it: Say $V$ is a vector space with an inner product above $\mathbb{C}$ (We don't know what ...
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1answer
79 views

Linear systems. Please help me solve this

Please help me solve this. Consider for every real number $a$ the linear system of equations: $$ \begin{align} x +( a + 1 )y + a^2 z &= a^3 \\ (1-a)x +( 1 - 2a )y &= a^3 \\ x +( a ...
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1answer
138 views

Proof: basis for Eigenspace with different Eigenvalue are independent

Question: Suppose we already have m linearly independent eigenvectors $\mu_1, \mu_2, ... \mu_m$. Say $A\mu_i = \lambda\mu_i$ for $i = 1, 2, ..., m$ where $\lambda_1, \lambda_2, ..., \lambda_m$ are ...
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0answers
37 views

An endomorphism sending a basis element to zero

Let $\mathbb R_n[X]$ be the vector space of polynomials of degree at most $n$. Let $u$ be the endomorphism $$u(P)=(X^2-1)P''-2XP'$$ I want to determine the determinant of $u$. So I proceed by ...
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1answer
36 views

Distinct Eigen values

What is the condition for a matrix to have distinct eigenvalues. We are working on large symmetric matrices where finding its characteristic equation is next to impossible. Still we need to ...
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1answer
57 views

Do zeros present along the diagonal yield complex eigenvalues?

I was told today by a friend that having a zero along there main diagonal of a matrix will promote complex eigenvalues. I do not believe this is true because the below matrix Z has a zero present ...
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votes
2answers
105 views

Must eigenvector matrix be invertible?

When reading eigenvector of a matrix, there is a formula: $AP = PD$ where in $P$, each column is A's eigenvector and $D$ is diagonal matrix with diagonal element being A's eigen values. Now coming ...
2
votes
1answer
27 views

relationships of symmetric matrices

I came across the following relationships, but I have no idea how to prove them. I would love to know they can be proved. Suppose $X$ and $Y$ are both symmetric matrices, relationship: $$(X + ...
2
votes
2answers
95 views

Calculating mean (Average) of vector

I'm trying to calculate PCA (Principle component analysis) and part of the equation is to calculate the mean of a vector $v$ and subtract each element of $v$ by it's mean. However, is this in column ...
1
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1answer
53 views

Prove one to one and onto of a linear transformation.

I have no idea how to proceed this. Thanks for your help!