# vector spaces

1. ### [Linear Alg] Determining which sets are subspaces of R[x]

1. Homework Statement Which of the following sets are subspaces of $R[x]?$ $W_1 = {f \in \mathbf R[x] : f(0) = 0}$ $W_2 = {f \in \mathbf R[x] : 2f(0) = f(1)}$ $W_3 = {f \in \mathbf R[x] : f(t) = f(1-t) \forall t \in \mathbf R}$ ##W_4 = {f \in \mathbf R[x] : f = \sum_{i=0}^n...
2. ### Determining if a subset W is a subspace of vector space V

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. 2. 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 ∈...
3. ### Is the zero Matrix a vector space?

1. Homework Statement So I have these two Matrices: M = \begin{pmatrix} a & -a-b \\ 0 & a \\ \end{pmatrix} and N = \begin{pmatrix} c & 0 \\ d & -c \\ \end{pmatrix} Where a,b,c,d ∈ ℝ Find a base for M, N, M +N and M ∩ N. 2. Homework Equations I know the 8 axioms about the vector spaces. 3...
4. ### Insights What Is a Tensor? - Comments

fresh_42 submitted a new PF Insights post What Is a Tensor? Continue reading the Original PF Insights Post.
5. ### I Understanding Vector Spaces with functions

Is the set of all differentiable functions ƒ:ℝ→ℝ such that ƒ'(0)=0 is a vector space over ℝ? I was given the answer yes by someone who is better at math than me and he tried to explain it to me, but I don't understand. I am having difficulty trying to conceptualize this idea of vector spaces...
6. ### I Prove the sequence is exact: 0 → ker(f) → V → im(f) → 0

Problem: Let f ∶ V → V be a linear operator on a finite-dimensional vector space V . Prove that the sequence 0 → ker(f) → V → im(f) → 0 is exact at each term. Attempt: If I call: a: 0 → ker(f), b: ker(f) → V, c: V → im(f), d: im(f) → 0. Then the sequence is exact at: ker(f) if...
7. ### Showing that Something is a Subspace of R^3

1. Homework Statement The question asks to show whether the following are sub-spaces of R^3. Here is the first problem. I want to make sure I'm on the right track. Problem: Show that W = {(x,y,z) : x,y,z ∈ ℝ; x = y + z} is a subspace of R^3. 2. Homework Equations None 3. The Attempt at a...
8. ### Modular arithmetic on vector spaces

1. Homework Statement Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W. 2. Homework Equations -Polynomial vector spaces...
9. ### Linear algebra: Prove the statement

1. Homework Statement Prove that \dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L) for every subspace \mathbb{F} and every linear transformation L of a vector space V of a finite dimension. 2. Homework Equations -Fundamental subspaces -Vector spaces 3. The Attempt at a Solution Theorem...
10. ### Linear algebra: Finding a basis for a space of polynomials

1. Homework Statement Let and are two basis of subspaces and http://www.sosmath.com/CBB/latexrender/pictures/69691c7bdcc3ce6d5d8a1361f22d04ac.png. [Broken] Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and...
11. ### Find a basis and dimension of a vector space

1. Homework Statement Find basis and dimension of V,W,V\cap W,V+W where V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\} 2. Homework Equations -Vector spaces 3. The Attempt at a Solution Could someone give a hint how to get general representation of...
12. ### Tangent spaces at different points on a manifold

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...
13. ### Help: All subspaces of 2x2 diagonal matrices

The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices. I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix. The answer given is here, relevant answer is (b): Imgur link: http://i.imgur.com/DKwt8cN.png...