# subspace

1. ### Invariant Subspace proof

I am having some trouble with part (ii) of the following question, part (i) is fine; Let V be a finite dimensional vector space and let f, g : V -> V be linear maps such that f∘g = id_V. Prove: (i) g∘f = id_V (ii) a subspace of V is f-invariant if and only if it is g-invariant I've tried the...

Hi, I hope someone can help. I'm trying to understand the following theorem (about subspaces): I understand the first property. However the second property doesn't make any sense to me. My professor said that the second property is saying that out of all the sets which have property (1)...
3. ### Prove that V is a subspace of Rn

Let V = {x E Rn | Ax= λx} where A is an n x n matrix and λ is a real scalar, together with the usual operations for vector addition and scalar multiplication from Rn. Prove that V is a subspace of Rn.
4. ### Subspace

Regarding the two subsets of R4: S = {(1, 1, 0, −1), (1, −2, 1, 6), (1, 1, 1, 0)}, T = {(x, y, z, x − 2y + z) : x, y, z ∈ R}. Show that T is a subspace of R4. Any help will be much appreciated. Do you know from where could I start? Thank you!
5. ### Two bases for this subspace

I need to give two bases for the following subspace. I'm able to find the spanning set, and thus, a possible basis. Would a second possible basis be the following: 2 0_____ 0 4_____ 0 0 0 -2____ 0 0_____ 3 0
6. ### Transformation problem

How can i prove this: X is the vector space generated by the functions sin ( x ) and cos ( x ) ( subspace of R functions in R ) . The derivative is a linear transformation on passing functions from space to space X to X. Find the D matrix representing the transformation on derivative with...
7. ### finding a basis of a subspace

I have been given the following S={(x,y,z,x+2y-z):x,y,z\in R I have been asked to show that this is a subspace of R4, find a basis for the subspace and write down the dimension of S. The first part is no problem I can show that this is a subspace by showing S contains the 0 vector and is...
8. ### Is A Subspace of a 3D Vector Space a 2D Plane?

Not sure how to elaborate on the question more than the title, so I'll just reiterate: Is A Subspace of a 3D Vector Space a 2D Plane? I'm just trying to visualize/understand this concept.
9. ### Proving Matrix (2x2) is a subspace.

Hi again, for this question I know that I have to prove that there is a zero vector in the matrix, but how do I do that?. Also I assumed that the matrix being the same structure after addition and scalar multiplication meant that it was sufficient proof (closure under addition and scalar...

11. ### Having difficulty with a subspace proof

Prove that V= {(x, y, x+y)} is a subspace of R^3. What is it? This is as far as I can get: 1) x=y=x+y=0 so 0ϵV 2) .... ​Any help is appreciated!
12. ### Finding an orthogonal basis

Hi all I have the following information: This question concerns the following two subsets of R3: S= {(2, 2, 2), (2, 3, 1)}, T= {(x, y, 2x − y) : x, y ∈ R} Find an orthogonal basis for T that includes the vector (2, 3, 1). I have a method for doing this but it's T that's...
13. ### subspace

Consider the circle in the xy-plane centered at the origin whose equation is x^2+y^2=1 Let W be the set of all vectors whose tail is at the origin and whose head is a point inside or on the circle. Is W a subspace of R^2 ? Explain.
14. ### Question about dimesions and basis of a subspace

So I'm working on a few homework problems and not really understand if a) my answer is correct and b) where the answers came from. I think it is because I do not fully understand the concepts of this part. So here a problem I was working on. Find a basis and calculate the dimension of the...
15. ### find a basis for the subspace P_2

Find a basis for the subspace $P_2$ spanned by the vectors $-2x^2+x-1,6x^2+3x+3,9$ they're the vectors $<-2,1,-1>, <6,3,3>, <0,0,9>$ I checked to make sure they are linearly independent and that they span $P_2$ so the basis is $\{<-2,1,-1>, <6,3,3>, <0,0,9>\}$ correct?
16. ### Subspace

Let $W$ be a subspace in $\mathbb{R^n}$ and $S=\{\vec{v} \in \mathbb{R^n} | \vec{w}\cdot\vec{v}=0, \forall \vec{w} \in W\}$ Prove $S$ is a subspace of $\mathbb{R^n}$ let $\vec{v_1}, \vec{v_2} \in S$ (i) $(\vec{v_1} + \vec{v_2})\cdot\vec{w}=\vec{v_1}\cdot\vec{w} + \vec{v_2}\cdot\vec{w}=0$...
17. ### Cylic Subspace

I have a question on cyclic subspaces. V is a finite dimensional space over the field F. There exists f:V->V It's given to me that , k is minimal, such that the set $(v,f(v),f^2(v),...,f^k(v))$ is dependent. Now I have to prove that 1-)$D=(v,f(v),f^2(v),...,f^(k-1))$ 2-)D spans Z(v,f) I...
18. ### Subspace Proof, finding a basis and dimension

Hey guys, I have a problem that I have to do, I am having a little trouble with it. Consider the set of matrices Show that W is a subspace ofM22. Find a basis for W, and hence find dim(W). (Make sure to demonstrate that your basis is linearly independent and spans W.) Thanks for your help.
19. ### Showing that a set spans the subspace of symmetric matrices

I'm not really sure how to show this. It's the part about symmetric matrices that throws me off. What I know: A symmetric matrix has the property that A = A^T. To show that the set spans, I could create a matrix and show that if there are leading ones in each row, without a pivot in the...
20. ### prove a vector subspace

Can someone please help me with this question, ? I know i first need to prove that it is empty, how do i show that the zero vector is in U? i know I have to take two elements of U and add them and see if I still am in the same space, but really not sure how to? any help appreciated.