Problem 13.5. Let V and W be inner product spaces and T є L(V : W). Let(..) v and (..)w denote their respective mner products. Let ui, , uk be an orthonormal basts o V and W1,…,wn an orthonormal bass...
Let (V,〈 , 〉v) and (W.〈 , 〉w) be finite-dimensional inner product spaces. Recall that the adjoint L* : W → V of a linear function L Hom(V,W) is completely determined by the equation <L(v), w/w,-(v, L* (w)של for every v є V and w є W . Use this to prove the following facts: (a) (Li + L2)* = Lİ + L: for Li, L26 Horn(V,W) (b) (α L)* =aL' for a R and L€ Horn(V,W) (c) (L*)* =...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> W be a linear transformation. (a) Prove that for every pair of ordered bases B = exists a unique m x n matrix A such that [T(E)]c = A[r3 for all e V. The matrix A is called the (B,C)-matrix of T, written A = c[T]b. (b) For each n E N, let Pm be the vector space of...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
Problem 3 (Inner Products). (a) Let V, W be two finite dimensional vector spaces, dim V = n, dim W-m and V x W-+ R be a bilinear function, i.e., for each a V and b E W: 1(a, r-Ay)-I(a,r) + λ|(a, y), for all r, y W, λ ε R and 1(u + λν, b)-1(u, b) + λ|(u, b), for all u, u ε ν, λ ε R. Thus for each fixed a E V, W 14-R is a...
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformations (a) Prove that if B-(Un . . . , v. . . . ,6) is a basis of V such that Bo-(Un .. . ,%) s a basis of ker(T) then (T(Fk+), , T(n)) is a basis of im(T) (b) Prove that if (w!, . . . ,u-, υ, . . . ,i)...