In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
Recall, a set ##X## is totally bounded if for each ##\epsilon>0##, there exists a finite number of open balls of radius ##\epsilon>0## that cover ##X##.
Question: How can I verify that the balls ##B(\epsilon j,\epsilon)## cover ##T##? In particular, why the condition ##\epsilon |j_i|\leq 2b##...
For this problem,
The solution for (a) is
I am slightly confused for ##p \in W## since I get ##a_3 = 2a_1## and ##a_2 = 2a_0##. Since ##a_3 = 2b##, ##a_2 = 2a##, ##a_1 = b##, ##a_0 = a##.
Anybody have this doubt too?
Kind wishes
Given a complex matrix ##A\in M_n(\mathbb{C})##, let ##X_A## be the subspace of ##M_n(\mathbb{C})## consisting of all the complex matrices ##M## commuting with ##A## (i.e., ##MA = AM##). Suppose ##A## has ##n## distinct eigenvalues. Find the dimension of ##X_A##.
Can a vector subspace have the same dimension as the space it is part of?
If so, can such a subspace have a Cartesian equation?
if so, can you give an example.
Thanks in advance;
Determine whether the following subsets U of M4x4is a subspace of the vector space V of all M4x4 matrices, with
the standard operations of matrix addition and scalar multiplication. If is not a subspace provide an example to
demonstrate a property that U does not possess.
a. The set U of all 4x4...
Hi all, I am a beginner in Linear Algebra. I am solving problems on vector spaces and subspaces from the book Introduction to Linear Algebra by Gilbert Strang. I have come across the following question:
Suppose P is a plane through (0,0,0) and L is a line through (0,0,0). The smallest vector...
Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##?
Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied...
Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks!
My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...
Let ##S## be a set of all polynomials of degree equal to or less than ##n## (n is fixed) and ##p(0)=p(1)##.
Then, a sample element of ##S## would look like:
$$
p(t) = c_0 + c_1t +c_2t^2 + \cdots + c_nt^n
$$
Now, to satisfy ##p(0)=p(1)## we must have
$$
\sum_{i=1}^{n} c_i =0
$$
What could...
I have a given point (vector) P in R^3 and a 2-dimensional linear subspace S (a plane) which consists of all elements of R^3 orthogonal to P.
The point P itself is element of S.
So I can write
P' ( x - P ) = 0
to characterize all such points x in R^3 orthogonal to P. P' means the transpose...
I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance.**My proof:** Take a limit point x of U that is not in U, but is in K (in other words x \in K \cap(\overline{U}-U))...
We only worry about finite vector spaces here.
I have been taught that a subspace ##W## of a vector space ##V## has a complementary subspace ##U## if ##V = U \oplus W##.
Besides, I understand that, given a finite vectorspace ##(\Bbb R, V, +)##, any subspace ##U## of ##V## has a complementary...
Ok, sorry, I am being lazy here. I am tutoring intro topology and doing some refreshers. Were given the subspace topology on [0,1] generated by intervals [a,b) and I need to answer whether under this topology, [0,1] is Hausdorff, Compact or Connected. I think my solutions work , but I am looking...
So the reason why I'm struggling with both of the problems is because I find vector spaces and subspaces hard to understand. I have read a lot, but I'm still confussed about these tasks.
1. So for problem 1, I can first tell you what I know about subspaces. I understand that a subspace is a...
Let ##n=\dim X## and ##m=\dim Y##.
Define a basis for ##X: y_1,...,y_m,z_{m+1},...,z_n##. The first ##m## terms are a basis for ##Y##. The remaining ##n-m## terms are a basis for its complement w.r.t ##X##. Let's call it ##Z##. ##X## is the direct sum of ##Y## and ##Z##; denote it as ##X=Y+Z##...
I am stuck on finding the dimension of the subspace. Here's what I have so far.
Proof: Let ##W = \lbrace x \in V : [x, y] = 0\rbrace##. We see ##[0, y] = 0##, so ##W## is non empty. Let ##u, v \in W## and ##\alpha, \beta## be scalars. Then ##[\alpha u + \beta v, y] = \alpha [u, y] + \beta [v...
Hey! :giggle: The three axioms for a subspace are:
S1. The set must be not-empty.
S2. The sum of two elements of the set must be contained in the set.
S3. The scalar product of each element of the set must be again in the set.
I have shown that:
- $\displaystyle{X_1=\left...
Problem:
Show that the set of differentiable real-valued functions ##f## on the interval ##(-4,4)## such that ##f'(-1) = 3f(2)## is a subspace of ##\mathbb{R}^{(-4,4)}##
This is my first bouts with rigorous mathematics and my brain is not at all wired for attacking problems like this (yet). I...
Let ##\mathscr{L_H}## be the usual lattice of subspaces of Hilbert space ##\mathscr{H}##, where for ##p,q\in\mathscr{H}## we write ##p\leq q## iff ##p## is a subspace of ##q##. Then, as discussed by, e.g., Beltrametti&Cassinelli https://books.google.com/books?id=yWoq_MRKAgcC&pg=PA98, this...
This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the...
Hey! 😊
Let $\mathbb{K}$ be a field and let $V$ be a $\mathbb{K}$-vector space.
Let $\phi,\psi:V\rightarrow V$ be linear maps, such that $\phi\circ\psi=\psi\circ\phi$.
I have shown using induction that if $U\leq_{\phi}V$ (i.e. it $U$ is a subspace and $\phi$-invariant), then...
Hey! 😊
Let $\mathbb{K}$ be a field and let $V$ a $\mathbb{K}$-vector space. Let $\phi, \psi:V\rightarrow V$ be linear operators, such that $\phi\circ\psi=\psi\circ\phi$.
Show that:
For $\lambda \in \text{spec}(\phi)$ it holds that $\text{Eig}(\phi, \lambda )\leq_{\psi}V$.
Let...
The proof that the set is a subspace is easy. What I don't get about this exercise is the dimension of the subspace. Why is the dimension of the subspace ##n-1##? I really don't have a clue on how to go through this.
I was learning about Degenerate Perturbation Theory and I encountered the term 'Degenerate Subspace', I didn't really understand what it meant so I came here to ask - what does it mean? will it matter if i'll say 'Degenerate space' instead of 'Degenerate Subspace'? and subspace of what? (...
Hey! :o
Let $1\leq m, n\in \mathbb{N}$, let $\phi :\mathbb{R}^n\rightarrow \mathbb{R}^m$ a linear map and let $U\leq_{\mathbb{R}}\mathbb{R}^n$, $W\leq_{\mathbb{R}}\mathbb{R}^m$ be subspaces.
I want to show that:
$\phi (U)$ is subspace of $\mathbb{R}^m$.
$\phi^{-1} (W)$ is subspace of...
I am assuming the set ##V## will have elements like the ones shown below.
## v_{1} = (200, 700, 2) ##
## v_{2} = (250, 800, 3) ##
...
1. What will be the vector space in this situation?
2. Would a subspace mean a subset of V with three or more bathrooms?
1. Let's show the three conditions for a subspace are satisfied:
Since ##\mathbf{0}\in \mathbb{R}^n##, ##A\times \mathbf{0} = \mathbf{0}\in S##.
Suppose ##x_1, x_2\in \mathbb{R}^n##, then ##A(x_1+x_2) = A(x_1)+A(x_2)\in S##.
Suppose ##x\in S## and ##\lambda\in \mathbb{R}##, then ##A(\lambda x) =...
Let's say we have n vectors in ℝ3. And say we have defined a subspace inside ℝ3 in the form of a sphere with radius r, and the center of the spheare is at P, where P is a vector in ℝ3.
What methods exists to find any linear combination of the n vectors, so that the sum of all of them, lies...
S is the set of solutions for the set of three equations...
x + (1 - a)y-1 + 2z + b2w = 0
ax + y - 3z + (a - a2)|w| = a3 - a
x + (a - b)y + z + 2a2w = b
I worked out...
The first equation is a subset of R4 when a = 1, b is any real.
The second equation is a subset of R4 when a = 1 or a = 0...
D is the set and the set contains the solutions to
x + (1 - m)y-1 + 2z + n2w = 0
I'm trying to find m, n values which means the set is a subspace of R (four dimensions).
===
Similarly, trying to find the m, n values that makes the following two expressions two separate subspaces, too.
mx +...
I had assumed that we had to put our values into a matrix so I did [1 2 -1 0; 1 -5 0 -1] and then I would do a=[1; 1] and repeat for b, c, and d. This is incorrect however. I also thought that it could be {(1, 2, -1, 0),(1, -5, 0, -1)} however this was not the answer, and I am unsure of what do...
Let ##\mathbb{V}## be a vector space and ##\mathbb{W}## be a subset of ##\mathbb{V}##, with the same operations.
Claim:
If ##\mathbb{W}## is non-empty, closed under addition and scalar multiplication, then ##\mathbb{W}## is a subspace of ##\mathbb{V}##.
A set is a vector space if it...
I want to exactly diagonalize the following Hamiltonian for ##10## number of sites and ##5## number of spinless fermions
$$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$
here ##L## is total number of sites, creation (##c^\dagger##) and...
Homework Statement
Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself,
and the lines through the origin. (Hint: Show that if W is a subspace of
##R^2## that contains two nonzero vectors lying along different lines through
the origin, then W must be all of...
Homework Statement
This is the exact phrasing form Linear Algebra Done Right by Axler:
Prove that the union of three subspaces of V is a subspace of V if and only if one of the subspaces contains the other two. [This exercise is surprisingly harder than the previous exercise, possibly because...
Homework Statement
"Let ##T## be a linear operator on a finite-dimensional vector space ##V## over an infinite field ##F##. Prove that ##T## is cyclic iff there are finitely many ##T##-invariant subspaces.
Homework Equations
T is a cyclic operator on V if: there exists a ##v\in V## such that...
Homework Statement
Let ##V## be the vector space of the sequences which take real values. Prove whether or not the following subsets ##W \in V## are subspaces of ##(V, +, \cdot)##
a) ## W = \{(a_n) \in V : \sum_{n=1}^\infty |a_n| < \infty\} ##
b) ## W = \{(a_n) \in V : \lim_{n\to \infty} a_n...
Homework Statement
Have to read a paper and somewhere along the line it claims that for any distinct ## \ket{\phi_{0}}## and ##\ket{\phi_{1}}## we can choose a basis s.t. ## \ket{\phi_{0}}= \cos\frac{\theta}{2}\ket{0} + \sin\frac{\theta}{2}\ket{1}, \hspace{0.5cm} \ket{\phi_{1}}=...
Homework Statement
Let V = RR be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V.
Homework Equations
W = {f ∈ V : f(1) = 1}
W = {f ∈ V: f(1) = 0}
W = {f ∈ V : ∃f ''(0)}
W = {f ∈ V: ∃f ''(x) ∀x ∈ R}
The...
Homework Statement
Find the dimension of the subspace of all vectors in ##\mathbb{R}^3## whose first and third entries are equal.
Homework EquationsThe Attempt at a Solution
So I arrived at two solutions and I'm not entirely sure which is the valid one.
#1
Let ##H \text{ be a subspace of }...
Homework Statement
From Linear Algebra and Its Applications, 5th Edition, David Lay
Chapter 4, Section 1, Question 32
Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...
Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.
I want to show that ##\mathbb{R}## is disconnected with the subspace topology. For this I considered that ##\mathbb{R} = \lim_{\delta n \longrightarrow 0 } (-\infty, n] \cup [n+\delta n, \infty)## and of course the intersection of these two open sets is empty.
What I'm not sure is about the...
Helo, I believe that the folowing exercise from Topology by Munkres is incorrect:
"Let A be a proper subset of X, and let B be a proper subsert of Y. If X and Y are conected, show that
##(X\times Y)-(A\times B)## is connected"
I think I can prove it wrong however I'm not sure and would like to...
Homework Statement
Determine the vector subspace generated by ##A = \{x^2 -x, 3 - x^2, 1+x \} \subset P^2(x)##
Homework EquationsThe Attempt at a Solution
I tried the usual check of vector addition and scalar multiplication to get the conditions that ##x## and ##y## should satisfy, but...
Homework Statement
Show that {(1, 2, 3), (3, 4, 5), (4, 5, 6)} does not span R3. Show that it spans the subspace of R3 consisting of all vectors lying in the plane with the equation x - 2y + z = 0.
Homework EquationsThe Attempt at a Solution
I made a matrix of:
A = [ 1 3 4 ; 2 4 5; 3 5 6] and...
Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions."
Regarding the two main operators, X and D, each has...