T X = R with standard topology { IneN} CR k= Show that the subspace topology on k is discrete.
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- Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.Let C2(-∞,∞)={f(x) in C(-∞,∞)|f''(x) exists for all x} be the set of differential functions. Show this is a subspace of C(-∞,∞).Let C[−π, π] be the vector space of functions that are continuous over the interval [−π, π]. Find the dimensionof the subspace of C[−π, π] that is spanned by the set {1, cos(2x), cos^2(x)}.
- If V is a vector space over F of dimension 5 and U and W are subspacesof V of dimension 3, prove that U ∩ W ≠ {0}. Generalize.In the vector space of all real-valued functions, find a basis for the subspace spanned by {sin t,sin 2t,sin t cost} .I have the subspace topology I am confused on part b and c specifically.
- Suppose U1; U2; :::; Um are Önite-dimensional subspaces of V .Prove thatU1 + U2 + ::: + Um is Önite-dimensionalanddim(U1 + U2 + ::: + Um) dim U1 + dim U2 + ::: + dim UmHow would I find whether there is a linear algebra subspace in R^3, inclusion of zero vector, closure under vector addition or scalar multiplication?Let V = { [x y z] in ℝ^3 ∶ z = 2x - y }. Is V a subspace of ℝ^3? If it is, what is the dimension of V? Any help (especially with details) would be greatly appreciated.
- Consider the subspace W of D, given by W = span(sin x, cos x). (a) Show that the differential operator D maps W into itself. (b) Find the matrix of D with respect to B = {sin x, cos x}. (c) Compute the derivative of f(x) = 3 sin x - 5 cos x indirectly and verify that it agrees with f'(x) as computed directly.can we have a polynomial Q(x,y,z) such that the subspace {(x,y,z):Q(x,y,z)=0} is a smooth submanifold diffeomorphic to T^2(torus) in step by step with out using chatgpt.How do you prove that W= im T when W is a T-invariant subspace, and V=ker + W. Where V is finite dimentional, when you let T be a element V.