# Subspace Definition and 20 Discussions

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.

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1. ### Prove that ##S## is a subspace of ##V##

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...
2. ### Engineering Signals & Systems with Linear Algebra

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...
3. ### I Can this work as a basis for S?

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...
4. ### Help with linear algebra: vectorspace and subspace

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...
5. ### A Is the "op" lattice ##\mathscr{L_H}^\perp## also atomistic....?

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...
6. ### I Closure in the subspace of linear combinations of vectors

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...
7. ### I Definition of a Degenerate Subspace (QM)

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? (...
8. ### I Finding a linear combination to enter a sphere

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...
9. ### How to show a subspace must be all of a vector space

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 ##R^2##)...
10. ### [Linear Algebra] Linear Transformations, Kernels and Ranges

Homework Statement Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range. a) ## T : ℝ → ℝ^2, T(x) = (x,x)## b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)## c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ## d) ## T: C[a,b] → ℝ, T(f) = f(a)## e) ##...
11. ### [Linear Algebra] Another question on subspaces

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...
12. ### I How to find admissible functions for a domain?

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...
13. ### Subspace question

Homework Statement Let X=ℝ3 and let V={(a,b,c) such that a2+b2=c2}. Is V a subspace of X? If so, what dimensions? Homework Equations A vector space V exists over a field F if V is an abelian group under addition, and if for each a ∈ F and v ∈ V, there is an element av ∈ V such that all of...
14. ### Linear algebra: Prove that the set is a subspace

Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...
15. ### How to determine the smallest subspace?

Two examples are: Given two subspaces ##U## and ##W## in a vector space ##V##, the smallest subspace in ##V## containing those two subspaces mentioned in the beginning is the sum between them, ##U+W##. The smallest subspace containing vectors in the list ##(u_1,u_2,...,u_n)## is...
16. ### Determining subspaces for all functions in a Vector space

Homework Statement First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0...
17. ### What subspace of 3x3 matrices is spanned by rank 1 matrices

So that's the question in the text. I having some issues I think with actually just comprehending what the question is asking me for. The texts answer is: all 3x3 matrices. My answer and reasoning is: the basis of the subspace of all rank 1 matrices is made up of the basis elements...
18. ### Subspace of a 4x4 matrix space

Homework Statement This question is taken from Linear Algebra Done Wrong by Treil. Question 7.5 of chapter 1 says this: What is the smallest subspace of the space of 4 4 matrices which contains all upper triangular matrices (aj,k = 0 for all j > k), and all symmetric matrices (A = AT )? What is...
19. ### Determine if all vectors of form (a,0,0) are subspace of R3

I have the feeling that it is, but I am not really sure how to start the proof. I know I have to prove both closure axioms; u,v ∈ W, u+v ∈ W and k∈ℝ and u∈W then ku ∈ W. Do I just pick a vector arbitrarily say a vector v = (x,y,z) and go from there?
20. ### Need help with proof of Vector Space (Ten Axioms)

Let S={x ∈ R; -π/2 < x < π/2 } and let V be the subset of R2 given by V=S^2={(x,y); -π/2 < x < π/2}, with vector addition ( (+) ). For each (for every) u ∈ V, For each (for every) v ∈ V with u=(x1 , y1) and v=(x2,y2) u+v = (arctan (tan(x1)+tan(x2)), arctan (tan(y1)+tan(y2)) ) Note: The...