Fix an element a E /V and ß Wv. Show that the 'product, aß defines a linear map
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator in V . Show that T has an adjoint, and describe T ∗ explicitly
4. Let L: V→ W be a linear map. Let w be an element of W. Let uo be an ele- ment of V such that LvO-w. Show that any solution of the equation L(X)-w is of type uo + u, where u is an element of the kernel of L.
QUESTION 8 Let (V,<,>) be an inner product space, and P: V – V a linear map. Choose the correct statement(s). Multiple choices might be correct and wrong choices have negative points. if P(V) = < W, V > Wand ||w|= 1, then P is an orthogonal projection. if P is an orthogonal projection, then < V- P(V), W> = 0 for any VEV, welmP. fW= Im P and {W 1,...,Wx} is an orthonormal basis for W then P(V) =...
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
Vectors pure and applied
Exercise 11.5.9 Let U and V be finite dimensional spaces over F and let θ : U linear map. v be a (i) Show that o is injective if and only if, given any finite dimensional vector space W map : V W such that over F and given any linear map α : U-+ W, there is a linear (ii) Show that θ is surjective if and only if, given any finite dimensional vector space...
1. Let F :V + V be a linear map, and let be a eigenvalue of F. Show that the set of all eigenvectors associated with is a subspace.
2/2 Problem 2 Suppose that the map T: D C R2R2 (u, v) T(u, v)- (TI(u, v), T2(u, v)) defines a change of variables whose Jacobian satisfies J(T) (u, v)1 for l (u, v) E D If R C D is a region whose area is 4, then what is the area of the region T(R) T(u, v)(u, v) E R? 5 marks
2/2 Problem 2 Suppose that the map T: D C R2R2 (u, v) T(u, v)- (TI(u, v),...
1. let V be a vector space and T an operator on V (i.e., a
linear map T: V--> V). Suppose that T^2 - 5T +6I = 0, where I is
the identity operator and 0 stands for the zero operator
...
Read Section 3.E and 3.F V) 1. Let V be a vector space and T an operator on V (i.e., a linear map T: V -» Suppose that T2 - 5T + 61 = 0, where I is...
Show that Ka,ß is a tree (a = 1 V B = 1)
4.5 Define Try/F, (a) a +a ++a for any a E F. The element TrF/F, (a) is called the trace of a with respect to the extension F4/F. (i) Show that Trm/F, (a) is an element of F4 for all a e F (ii) Show that the map TrF F, (a) Tr/F, F F4 is an F-linear transformation, where both Fm and F are viewed as Fg vector spaces over (iii) Show that Tr/F, is surjective (iv) Let BE F....