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 ...

learn more… | top users | synonyms

1
vote
1answer
83 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
0
votes
3answers
21 views

Find a matrix which maximizes expression

Assume I have column vectors $x,y\in\mathbb{R}^n$, and the following expression $$ A\in M_n(\mathbb{R}),\ |\det A|\leq 1,\ K(A) = x^tAy $$ How can I find the matrix $A$ that maximizes expression ...
1
vote
1answer
26 views

Isomorphism between $E$ and $E^*$

Show that there does not exist a isomorphism $\phi:E\rightarrow E^*$ that it takes every basis of $E$ to its dual basis. ($E$ is a vector space over field K and $\text{dim}E=n$ .) My attempt: There ...
-4
votes
0answers
27 views

Proving linear algebra equation

I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent. I would appreciate any help on this. Prove A = ...
1
vote
1answer
29 views

Reformat this math formula to not need parenthesis

I've got a pricing equation that I am working with for an eCommerce site and I need to reformat this equation to not need parenthesis. Original Formula: {price} + ({length} * ({ppf} + ...
1
vote
1answer
28 views

Finding the standard basis for $\mathbb{R}^4$ that can be added to the set $\{(1,-4,2,-3),(-3,8,-4,6)\}$ to produce a basis for $\mathbb{R}^4$

Finding the standard basis for $\mathbb{R}^4$ that can be added to the set $\{(1,-4,2,-3),(-3,8,-4,6)\}$ to produce a basis for $\mathbb{R}^4$. I first check that the two vectors in the set are not ...
3
votes
1answer
42 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
2
votes
0answers
16 views

Pullback maps and an equallity

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
0
votes
0answers
20 views

Solving an specific equation involving cos and sin

Here is the equation: $|a|\sin(\alpha+2\theta)+|b|\sin(\beta+\theta)=0$, where $\theta$ is the variable, $a$ and $b$ are complex constant and their corresponding arguments are $\alpha$ and $\beta$. ...
0
votes
3answers
42 views

Finding the Characteristic Equation

For the following matrix I need to find $$\begin{bmatrix}-3 & 2 &1 \\3 & -4 & -3 \\-8 & 8 & 6 \end{bmatrix}$$ a. Characteristic Polynomial of $A$ b. Eigen Values c. Eigen ...
-2
votes
1answer
30 views

Orthogonal vectors and potential

given the potential $ψ(x;y)$, such that $dψ=−u_2dx+u_1dy$, why are $∇ψ=(−u_2;u_1)$ and $ψ(x;y)=c$ orthogonal vectors ? $c \in \mathbb{R}$ is a constant, and $\mathbf{u}(x; y) = (u_1(x;y); u_2(x;y))$, ...
-2
votes
2answers
24 views

Prove $\exists$ $v \in V$ so that $(v , f(v))$ is a basis of $V$

maybe you guys can help me with this one. Let's say we have a vector space $V$ with $dim(V) = 2$ and we have a linear map $f : V \rightarrow V$ with $f^2 := f \circ f = 0$ ...
0
votes
2answers
31 views

How to determine if matrices are similar?

Trying to teach myself some Linear Algebra, now trying to study about similar matrices concept, but i am having some trouble (maybe because i am trying to teach myself), found a question online and i ...
0
votes
1answer
21 views

Sets of compositions of homomorphisms

I am looking of a relation in the form: $$ Hom(X,Z) = Hom(X,Y)\otimes Hom(Y,Z), $$ or: $$ Hom(X,Z) \subseteq Hom(X,Y)\otimes Hom(Y,Z), $$ or similar (maybe it's not a tensor product? maybe the ...
0
votes
2answers
23 views

Proof equivalence relation

It is given the set $A := \left\{ n \in \mathbb{N} : n \le N, \; N \ne 0 \right\}$. For $a,\,b \in A$ we place $a\text{R}b$ if and only if $a,\,N$ have the same least common multiple of $b,\,N$. Prove ...
0
votes
0answers
9 views

Linear order question?

I am working on some homework and can someone assist me with these linear order problems? 1.Given a linear order ≤ on Π define s[U,V] ( U,V∈Π ) as the set of all X∈Π such that U≤X≤V or V≤X≤U . Show ...
4
votes
3answers
409 views

Unusual result to the addition

Question: Prove that (666... to n digits)^2 + (888... to n digits)=(444... to 2n digits) My way: I just proved the given equation for three values of n and written at the bottom. "Since the ...
1
vote
0answers
45 views

A condition for surjectivity of a linear map

Let $V,W$ be vector spaces (not necessarily finite dimensional!), and let $W^*$ the dual of $W$. Let $$A:V\longrightarrow W^*$$ be a linear map. What conditions do I have to put on $V$ and especially ...
1
vote
0answers
22 views

One dimensional space $(\Lambda ^n V)^{1/2}$

Let $V$ be an $n$ dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ on $V\oplus V^*$ such that $(v+\xi , ...
0
votes
0answers
31 views

a matrix metric

Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ matrices of the same size. We'll denote $E(U_i,V_i)= \max_v \, |(U_i -V_i)v|$ (max over all the quantum states), $U=\prod_i U_i$ and $V=\prod_i ...
2
votes
0answers
46 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
5
votes
2answers
92 views

Eigenvalues of linear operator $F(A) = AB + BA$

Let $B$ be the $n \times n$ square matrix; $\lambda_1, \lambda_2, \dots, \lambda_n$ are its pairwise distinct eigenvalues. For all $n \times n$ matrix $A$ let me define $F(A) = AB + BA$. We can ...
-1
votes
2answers
17 views

graph quadratic form and find the equation of asymptotes

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes. $$ \left\{ \begin{aligned} ...
0
votes
1answer
31 views

prove that there exists an upper triangular matrix U such that (U^T)U=A

Let A be a positive definite matrix \begin{pmatrix} a & b \\ b & c \\ \end{pmatrix} prove that there exists an upper triangular matrix U such that U transpose times U equals A. I'm ...
0
votes
1answer
12 views

What is the orthogonal complement of three linearly independent vectors in the 3-dimensional space?

If I have 3 linearly independent vectors, assume the standard basis, in R3, what would be its orthogonal complement? Would there even be one. Isn't the entire space represented by the standard basis?
2
votes
1answer
23 views

Matrix $B \in M_n(S)$, for $S$ an $R$-algebra, with $R$-independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$-independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I ...
0
votes
1answer
33 views

Rapid way to prove $ [e_{ij},e_{lk}]=\delta_{jl}e_{ik}-\delta_{ki}e_{lj} $

Let $e_{ij}$ denote the $n\times n$ matrix with entries all zero but the $(i,j)$th one, in which we put $1$. Let then $\delta_{ij}$ be the Kronecker Delta. Finally $[A,B]:=AB-BA$ is the commutator ...
0
votes
1answer
15 views

Find if a form is symmetric or skew-symmetric

Consider the set of all n × n matrices in R. Given the defined function Φ: $M$(n,n)× $M$(n,n) → R , which Φ(A,B) = $tr$(A$^T$JB) , where J is a skew-symmetric n × n matrix , define if Φ is a ...
0
votes
2answers
37 views

Consider the vector space V = {(a, 1 + a) | a ∈ R} with irregular definitions of addition and multiplication

with addition and scalar multiplication defined by (a, 1 + a) ⊕ (b, 1 + b) = (a + b, 1 + a + b) k '*' (a, 1 + a) = (ka, 1 + ka), k ∈ R find a basis for V. I started off with taking the general ...
3
votes
1answer
43 views

Calculating the rank of a Boolean matrix and Boolean matrix factorization

I am interesting in some sort of algorithm for calculating the Boolean rank of small $M \times N$ Boolean matrices. Just to be clear, by Boolean matrices I mean matrices with entries $0$ or $1$ where ...
1
vote
2answers
26 views

Orthonormal basis for the null space of almost-Householder matrix

A matrix $H$ is defined as: $$H = I - vv^T$$ where $v$ is a unit vector. What is the rank of $H$? What would be an orthonormal basis for the null space of $H$? How do we find the number of zero ...
-3
votes
1answer
37 views

Word problem to help me in my math class [on hold]

an estate valued at 124,104 is to be divided between two sons so that the older son receives twice as much as the younger son find each sons share of the estate
1
vote
3answers
33 views

Eigenvalues of Householder matrix

What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it to me intuitively or with a simple proof?
1
vote
0answers
43 views

How can you do algebra with rounded numbers?

I have a series of seemingly simple algebra problems: 9*x = 5, 5*x = 4, 4*x = 3, 1*x = 1 and ...
0
votes
1answer
23 views

Determine the dimension of $U+W$ and of $U \cap W$. Which sums are direct sums?

Problem: Determine the dimension of the sum $U + W$ and of the intersection $U \cap W$ of the following subspaces $U$ and $W$. Which sums are direct sums? 1) $U = \text{span}\left\{(1,1,1)\right\}$ ...
0
votes
0answers
35 views

What is the difference between the scalar and vector components of a vector?

What is a scalar component of a vector and what is a vector component of a vector. suppose a vector is making and angle theta with the origin then in my book it is written that its x component is the ...
1
vote
0answers
30 views

Functions linearly independent and linearly independent gradients?

Let $F_1,...,F_n: \mathbb{R}^n \rightarrow \mathbb{R}$ be a set of $C^{1}$ functions. Is it true that they are linearly independent on a joint level set $\Omega:= \{ p \in \mathbb{R}^n; ...
0
votes
0answers
14 views

Relation between bilinear symplectic forms and symplectic matrices

1. Symplectic Forms Let $F : \mathbb{K}^{2n} \times \mathbb{K}^{2n} \to \mathbb{K}$ be a bilinear skew-symmetric nondegenerate form (as known as symplectic form). Then $F(u,v) = u^TAv$ where $A = ...
1
vote
1answer
26 views

Curl: invariant under change of basis or not?

I wondered how the curl$$\text{rot}\mathbf{F}=\left( \begin{array}{ccc}\partial_y F_3-\partial_z F_2 \\ \partial_z F_1-\partial_x F_3 \\ \partial_x F_2-\partial_y F_1 \end{array} \right)$$of a vector ...
0
votes
1answer
29 views

Why the Householder matrix is orthogonal?

A Householder matrix $H = I - c u u^T$, where $c$ is a constant and $u$ is a unit vector, always comes out orthogonal and full rank. Why $H$ is orthogonal (looking for an intuitive proof rather than ...
1
vote
0answers
25 views

Question about the coordinates in a new origin on the plane.

I'm reading a book on analytic geometry, specifically on a chapter on change of coordinates. It says that having the origin $O$, one point $P$ and a new origin $O'$, the vector that describes the ...
0
votes
3answers
30 views

Show that $\ker \hat{T} = \text{ann}(\text{range } T)$

This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by ...
-1
votes
3answers
49 views

Representing a vector in $\mathbb{R}^{3}$ as sum of only two vectors in $\mathbb{R}^{3}$

Is it possible? Or more generally can any vector in $\mathbb{R}^{n}$ can be represented as sum of (n-1) or less vectors in $\mathbb{R}^{n}$? -----EDIT----- What I basically want to ask is that can ...
0
votes
0answers
32 views

Best way to quantify the difference between two vectors

There are plenty of ways of showing an error, or rather a deviation, between two vector quantities. What is the best choice? Specifically, at every timestep, I am comparing two vectors of curvature ...
2
votes
2answers
28 views

A theorem of symmetric positive definite matrix.

Is the following true? Let $g=(g_{ij})\in M(n,\Bbb R)$ be a symmetric positive-definite matrix and let $a=(a_1,\ldots,a_n)\in\Bbb R^n$ be any vector. Then, $$v^Tgv=1\implies (v\cdot a)^2\leq ...
2
votes
1answer
25 views

Is convex hull linear subspace of linear hull?

We have some convex and compact supset $G$ of banach space $B$ and finitely many points ${x_1,...,x_N}$ . The question is : does the convex hull $C$ of ${x_1,...,x_N}$ a linear subspace of space ...
1
vote
1answer
52 views

Does $A$ and $(A+I)^{-1}$ commute for positive operator $A$?

Suppose that $A$ is a bounded positive operator ($A \geqslant 0$) on some Hilbert space. Can I say that $A$ and $(A+I)^{-1}$ commute?
0
votes
0answers
24 views

Matrix representation of another matrix

Let $\mathbf{c}\in \mathbb{R}^n$ and $\mathbf{X}(s)= \begin{bmatrix} X_{11}(s) & X_{12}(s) & \cdots & X_{1n}(s) \\ X_{21}(s) & X_{22}(s) & \cdots & X_{2n}(s) \\ \vdots & ...
0
votes
0answers
21 views

Compressive sensing for complex matrix

I'm fairly new to compressive sensing, and I have been looking for a MATLAB implementation of the problem $$ A x = b $$ where $A$ is non square, $x$ is kind of sparse and all the numbers involved are ...
2
votes
2answers
17 views

For certain positive semidefinite matrices, subtracting the outer product of their row-sums does not change the positive semidefiniteness

Let $e$ denote the vector of all ones, $J=ee^T$ and $\langle A,B\rangle = trace(AB^T)$. Consider a symmetric positive semidefinite (psd) matrix $A\geq 0$ (that is, $a_{ij} \geq 0$ for all entries) ...