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)

3
votes
2answers
73 views

Under what conditions on the matrices $A$ and $B$ does $AXA=B$ have infinitely many solutions $X$?

The only things I can conclude are the dimensions of the matrices and the fact that $A$ is singular since otherwise only one solution would exist. I have no idea where to go next...
0
votes
1answer
64 views

project a point onto the intersection of surfaces

I have several non linear equations $g_i$ that represent surfaces $s_i$. Their intersection form the surface $S$. For example $s_1 : g_1(x_1,x_2,...,x_n)=c_1$ ... $s_n : g_m(x_1,x_2,...,x_n)=c_m$ ...
1
vote
2answers
35 views

How to define a cloud of points relative to a vector path?

I've been researching and playing with examples of particle clouds in a graphics visualization. Most use shape geometries to define a field of particles, or parameters for distributing them randomly ...
3
votes
1answer
95 views

Would the author most certainly be talking about a vector space over $\mathbb{R}$/$\mathbb{C}$ here?

I am working on the following problem in Serg Lang's Linear Algebra book: In the vector space of functions, what is the function satisfying the condition VS2? For reference VS2 is: There is ...
3
votes
3answers
99 views

Show surjectivity of a linear map

It pains me to say that this bewilders me, but here's the problem. All I want to do is show that: Given $T$ a linear operator on some finite-dimensional space $V$, with the property that $Im(T) = ...
1
vote
2answers
67 views

Finding the basis for null space of the matrix $[0^T, 1^T]$

As part of some work I am doing, I am looking into the basis for the null space of the following matrix $$ [\mathbf{0}^T, \mathbf{1}^T], $$ where the vector of zeros is of length n and the vector of ...
5
votes
3answers
2k views

Trouble understanding Sum of Subspaces

I started reviewing linear algebra, from a different textbook (Axler's), after taking a fast paced summer class. Unfortunately, I've become confused with a concept that is introduced at the end of ...
1
vote
1answer
142 views

If two functionals have the same kernel, then one is a multiple of the other [duplicate]

I would like some help with this exercise. Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$. I expect your suggestions. ...
5
votes
1answer
218 views

Inequality involving traces and matrix inversions

The following question kept me wondering for some weeks: Given the symmetric matrices $A,B,C\in\mathbb{R}^{n\times n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
1
vote
0answers
185 views

What can be the possible rank of adjoint of matrix of order n? [duplicate]

Let $ A $ be matrix of order $ n $. What may the possible ranks of $\mathop{\rm adj} (A) $? I think the possible answers are $0$, $1$, and $n$.
1
vote
1answer
52 views

Restricting solution $x$ to $Ax = b$ to natural numbers

Suppose that $A$ is $(n-1) \times n$ matrix that consists only of natural numbers (that is 0 and positive integers.) $b$ is $(n-1) \times 1$ matrix (vector) that only consists of natural numbers. For ...
2
votes
2answers
132 views

Prove that the determinants are equal

$$ Let\ A= \begin{bmatrix} 0 & a^2 & b^2 & c^2\\ a^2 & 0 & z^2 & y^2\\ b^2 & z^2 & 0 & x^2\\ c^2 & y^2 & x^2 & ...
1
vote
1answer
280 views

Odd town Even town explanation.

I am struggling to understand the solution to the following problem: If $\mathcal F\subset 2^{[n]}$ such that for each $F_1$ and $F_2$ in $\mathcal F$ we have $|F_1|,|F_2|\equiv 1 \bmod 2$ and ...
-1
votes
2answers
37 views

How to find rank of linear operator T on inner product space

Please see my question at this page. Now let $V=\mathbb{C}^n$. What is the rank of linear operator $T$? and what about its eigenvalues?
3
votes
2answers
150 views

Find Intersection Of 2 Sub-spaces

$U=span((1,1 ,0),(2 ,0 ,1))$ $W=span((1,1,1),(5,3,1))$ what is $U \cap W$? can I find the rref of them both and to find the intersections? that mean $U= \begin{pmatrix} 1 & 0 & 0.5 \\ 0 ...
2
votes
1answer
640 views

How to find adjoint of linear operator T on inner product space V

Let $V$ be an inner product space and $T$ a linear operator with $T(\alpha) = (\alpha,\beta)\gamma$ for fixed elements $\beta,\gamma \in V$. I now that $T$ is linear operator. How we can show that ...
0
votes
1answer
75 views

If Span(A)=Span(B) then $A \cap B \neq \emptyset$

Let the be V a vector space and $A,B \subset V$ sub-sets of V. If Span(A)=Span(B) then $A \cap B \neq \emptyset$ What I thought is that Span(A)=Span(B) mean that for all $a \in A$,$b \in B$ ...
3
votes
1answer
51 views

Find some informations about a Space

Let $W=\{(x,y,z):x-2y+z=0\}$ I wanna find: The basis of W Projection of u=(1,1,2) over W The dimension of W The orthogonal complement of W ($W^{\bot}$) The dimension of $W^{\bot}$ Basis of ...
0
votes
1answer
251 views

Finding a parabola from three points algebraically

I'm looking for a method to solve for $A$, $B$, and $C$ in the equation for a parabola given three points $(x_i, y_i), i = 1, 2, 3$. To start, I thought I should try to solve the system of equations: ...
0
votes
1answer
62 views

Killing form and Roots

I know that the roots of a Lie Algebra are functionals such that if $\alpha$ is a root and $h \in \mathfrak h$ is an element of the Cartan subalgebra, then $\alpha(h)$ is an eigenvalue. I'm looking ...
0
votes
1answer
40 views

Adjoint of Generalized Eigenvalue Problem

I have seen that for a given matrix A, its adjoint A* is defined as the conjugate transpose of A and has eigenvalues that are conjugates of the eigenvalues for A. I was wondering if there is a similar ...
0
votes
1answer
37 views

Intersection of two lines, computing the t-value

I have two points $(x_1,y_1)$ and $(x_2,y_2)$ that define my line, and also I have the other line in intercept slope form, $y=mx+b$, I know that these two lines intersect, so, I was wondering how to ...
4
votes
3answers
978 views

Interior product between differential forms and vector fields

I don't understand what is meant when someone writes that forms (or form fields) "eat" vectors (or vector fields). For example when I have a one form field ω=3dx+5dy+3xdz and a vector field ...
0
votes
3answers
57 views

The set of all Null linear maps is a subspace.

Suppose V and W are finite dimensional. Let $v \in V$. Let $E = {T \in L(V, W): Tv = 0}$ where L(V, W) is the set of all linear transformations from a vector space V to a vector space W. Show that E ...
1
vote
1answer
112 views

Inclusion induce an isomorphism

I have a small question If $E,F$ are two sub-modules of a module $G$, how to prove that the inclusion $E \rightarrow E+F$ induces an isomorphism $(E/E \cap F) \simeq (E + F) / F$ Please Thank you ...
5
votes
0answers
256 views

Proof of rank-nullity via the first isomorphism theorem

I was thinking about the proof of the rank-nullity theorem and I thought about proving it as follows. I just wondered whether this proof worked? Lemma. If $V$ is a finite-dimensional $F$-vector space ...
3
votes
1answer
224 views

Dominant eigenvector by looking at rows of matrix raised to a power

I'm not strong in linear algebra. I encountered this thing and being curious I want to know a bit more about it. I'm playing with 3x3 real valued matrices in some graphics application, I'm developing. ...
0
votes
1answer
60 views

minimizing sum of different least squares?

Can we write the minimization problem: $$\operatorname{min}\limits_{x\in\mathbb{R}^n}\sum_{i=1}^{n}\|C_i x-b_i\|_2^2$$ as a least square problem?
0
votes
1answer
65 views

If $A$ is an invertible matrix, then the column span of $A$ equals $F^{n\times1}$

If $A\in\mathbb{F}^{n\times n}$ is invertible so $\mathbb{F}^{n \times 1}=\text{span}\{C_i(A)\mid1\leq i \leq n\}$ What does being invertible has to do with the column span (unlike any other matrix)? ...
1
vote
2answers
60 views

Solving Multiple Equations with Many Variables

Here's a problem I have stumbled upon, which may have a straightforward solution with linear algebra. If so, I cannot see it. Choose $n > 0 \in \mathbb N$, and consider the sequence of equations: ...
1
vote
2answers
193 views

Scalar-by-matrix Derivative of Quadratic Product

I'd like to know $\frac{\partial f(\mathbf{U})}{\partial \mathbf{U}}$, i.e., the 'by-matrix derivative' of the following scalar function $f(\mathbf{U})$ w.r.t. $\mathbf{U}$. $$f(\mathbf{U}) = ...
1
vote
1answer
29 views

A functional is a multiple of another

In halmos book there's a question that says: If $y $ and $z$ are linear functionals on the same vector space and $[x, y]=0$* whenever $[x, z]=0$ then show that there exists a scalar $\alpha$ such ...
1
vote
1answer
204 views

If the union of linear spans is the span of unions, then one of two spans is a subset of the other

If $span(A) \cup span(B)=span(A\cup B)$, then $span(A) \subseteq span(B)$ or $span(B)\subseteq span(A)$. I can not see way this claim is right
1
vote
0answers
181 views

Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
1
vote
0answers
398 views

Calculate 3D-coordinates of a cube's points from the points on the projections

I have a following optical system: 3 cams (left and top, which is orthogonal to the left, and right, which is parallel to the left and orthogonal to the top) and the 2 cubes in the 3D-space with ...
0
votes
1answer
289 views

How can I refer a 3D pose (position + orientation) to a different coordinate system?

I'm working on a robotics project where all poses and marker positions/orientations are stored as a matrix: $$ \mathbf{P} =\begin{bmatrix} \mathbf{R} & \mathbf{t}\\ ...
1
vote
1answer
45 views

For what k is $\mathcal{M}_{m \times n}$ isomorphic to $\mathbb{R}^{k}$?

For what k is $\mathcal{M}_{m \times n}$ isomorphic to $\mathbb{R}^{k}$ ? I get a feel but am unable to prove it.
3
votes
2answers
144 views

Transforming $2D$ coordinates

Lets say from coordinate system 1, we have 3 points which consists of a triangle. The vertices are located at $(50,120) , (70,150) , (100,100)$. Now, coordinate system 2 consists also of a triangle, ...
0
votes
1answer
303 views

differences and similarities between Linear transformations, Linear functionals, Dual Spaces and Isomorphisms

Can someone please tell me the exact differences and similarities between Linear transformations, Linear functionals, Dual Spaces and Isomorphisms? I am very confused. I would appreciate if you can ...
15
votes
1answer
454 views

Every endomorphims is a linear combination of how many idempotents in infinite dimensions?

Every endomorphism of a finite-dimensional vector space is a linear combination of at most three idempotents, and the constant three is best possible, as Clément de Seguins has shown in this paper. ...
1
vote
1answer
41 views

Intersection between 2 lines

This problem is related to 3d perspective clipping, in other words, clipping a polygon against a truncated pyramided (frustrum), the author uses similar triangles to define the slope of each plane ...
1
vote
1answer
59 views

Showing a matrix is nilpotent if its charateristic polynomial is $t^n$ mod ${\rm nil}(R)$

Let $R$ be a commutative ring. How to prove the following: If $\chi_A(t) \equiv t^n \bmod\operatorname{nil}(R)$ then $A \in M_n(R)$ is nilpotent. Note $\chi_A$ is the characteristic polynomial ...
4
votes
2answers
114 views

Prove that $a^3 + 2b^3 + 4c^3 − 6abc \neq 0$ if at least one of $a$, $b$, and $c$ are non-zero [closed]

Prove for $a, b, c \in \mathbb Q$ that $a^3 + 2b^3 + 4c^3 − 6abc \neq 0$ if at least one of $a$, $b$, and $c$ are non-zero without resorting to field theory or linear algebra.
0
votes
1answer
121 views

The Matrix of the Differentiation map.

Suppose $T \in L(P_3(R), P_2(R))$ is the differentiation map defined by $Tp = p'$. Find a basis of $P_3(R)$ and a basis of $P_2(R)$ such that the matrix of T with respect to these bases is $ \left( ...
2
votes
1answer
175 views

Is trace of the product of two p.s.d. matrix always nonnegative?

Given $X\in S_+$ and $Y \in S_+$ Is $tr(XY)\geq 0$ for all $X$ and $Y$?
2
votes
2answers
37 views

Proving a matrix identity: if $Z = VT^{-1}V^T$, then $ (I - Z + Z(I + Z)^{-1}Z)^{-1} = I + Z $

I'm walking through a least squares derivation for the Kalman Filter, and after several hours I'm still unable to derive the statement made on page 15. In particular, that for a matrix $Z = ...
1
vote
1answer
69 views

Intersection of two lines in 3D

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. I want to find the coordiantes of the point $C=(x,y,z) $. The line segments $AC$ and $BC$ make equal angle $\alpha$ with ...
0
votes
0answers
36 views

Linear Algebra-Intersections and Distances in $\mathbb{R}^3$ and $\mathbb{R}^2$.

Self-study here. I need to check my answers. What is the relationship between the line $x=x_0+at,y=y_0+bt, z=z_0+ct$ and the plane $ax+by+cz=0$? Explain. T or F. If $a, b$ and $c$ are not all zero, ...
2
votes
2answers
143 views

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
2
votes
1answer
155 views

Is there vector space $\mathbb R $ over $\mathbb R $ with dimension $m\neq1$?

I am searching a new scalar product that with this scalar product the vector space $\mathbb R $over $\mathbb R $ with ordinary vector sum has dimension $m$ , which $m$ is any arbitrary natural ...