3.50. A closed linear subspace M of a Hilbert space H is invariant under an operator T if and only if M¹ is invariant under T.
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- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).For which of the following pairs is the set Ω not a basis for the vector subspace U ≤ R2[x]?4. Consider the following subspaces of P.H = Span{1 + t, 1 − t3} and G = Span{1 + t + t2, t − t3, 1 + t + t3}Find dim H, dim G and dim H ∩ G.
- Determine whether the following are subspaces of C[−1, 1]: The set of continuous nondecreasing functions on [−1, 1]Is the set M={1/n | n ε z+} compact as a subspace of R?4. Consider the following subspaces of P. H = Span{1+t, 1-t3 } and G = Span{1+t+t 2, t – t 3, 1+t+t 3} Find dim H, dim G and dim H intersection G.
- In P2 consider the subspace H = Span {f(x), g(x), h(x)} where f (x) = x2 + 3, g(x) = x + 1, and h(x) = 2x2 −3x + 3 a) Give 3 other elements in H.Note: Be certain to indicate how you selected your elements of choice. b) Determine if the set {f (x), g(x), h(x)} is linearly independent.Suppose V is finite-dimensional and S;T 2 L.V /. Prove that ST isinvertible if and only if both S and T are invertibleLet T: V-->W be a linear transformation between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let A= [ T]C<--B· Show that nullity(T) = nullity(A).
- 1. In each item if W is a real vector subspace of V: a) V= ℳ 2 (IR) and W = {A ∈ V | A is not invertible}. If not, justify! b) V = IR4 and W = {(x, y, z, t) ∈ IR4 | z is an integer}. If not, justify!For each of the following linear operators T on the vector space V,determine whether the given subspace W is a T-invariant subspace ofv.consider the subspace W of D given by W = span(e^2x, e^-2x) show that the differential operator D maps W into itself compute f(x) = cos(x) + 2xcos(x)