# What is Subspace: Definition and 570 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. ### Vector Subspaces: Determining U as a Subspace of M4x4 Matrices

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...
2. ### I Smallest subspace if a plane and a line are passing through the origin

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...
3. ### 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...
4. ### 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...
5. ### 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...
6. ### MHB Basis of linear subspace

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...
7. ### Proof of Subspace Topology Problem: Error Identification & Explanation

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...
8. ### Finding a complementary subspace ##U## | Linear Algebra

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...
9. ### I Subspace Topology Basics

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...
10. ### 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...
11. ### Cartesian sum of subspace and quotient space isomorphic to whole space

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##...
12. ### Finding the dimension of a subspace

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...