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 ∈...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...