that is Give an example of a linear transformation T: R2X2 → Ral surjective (onto), however is not one-to-one. O No linear transformation, T, exists which satisfies these conditions. There exist such a linear transformation, T. For example: sat
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
(1 point) Find the orthogonal projection of U = onto the subspace W of R4 spanned by --0-0-1 Uw =
(1 point) Find the orthogonal projection of 11 onto the subspace W of R4 spanned by 1 2 -2 and *20 -2 projw() =
(1 point) Find the orthogonal projection of V = onto the subspace V of R4 spanned by X1 = and X2 = 3/2 projv(v) = -39/2
Let T: R4 → R3 be the linear transformation represented by T(x) = Ax, where 1 A = 0 -2 1 0 1 2 3 . 0 0 1 0 (a) Find the dimension of the domain. (b) Find the dimension of the range. (C) Find the dimension of the kernel. (d) Is T one-to-one? Explain. O T is one-to-one since the ker(T) = {0}. O T is one-to-one since the ker(T) = {0}. O T is not one-to-one since...
(1 point) Find the orthogonal projection of 4 -11 11 onto the subspace W of R4 spanned by 2 2 2 -3 and 1 2 projw(1) =
(1 point) Find the orthogonal projection of -1 -5 V = 9 -11 onto the subspace V of R4 spanned by -4 -2 -4. -5 X1 = and X2 1 -28 -4 0 projv(v)
A linear mapping Φ from one vector space to another is one-to-one and is onto. Св.V Rn is an isomorphism. Show that the coordinate map Св is an isomorphism from B to R. (Show by proofs of contradiction.)