Assignment 3: Problem 1. a. Give the definition of a subspace of a vector space V. (One must use the lecture not b. State the Theorem 1 of Week 6 which is known as the "Subspace criteria." (You do to include the proof of it). (One must use the lecture notes). c. Use the above Theorem to show that the following subset of R3 forms a subspace of W = {(x1,x2, #3) R$ : 4x1 + 3x2 - 5x3 = 0}. Problem 2. a. State the definition of a basis for a vector space and the definition of the dimension of space. (One must use the lecture notes). b. State the definition of the "coordinates of a vector v relative to a basis B. (One must lecture notes) c. Consider the following subspace of M2x2(R): C = {[ 2a W 3b a + b 1. Find a basis B for W and determine dim(W). ii. Find the coordinate vector us relative to the basis B you found in part (1) o] M2x2(R): a, b, c R} 97 Hints: (a) Problem 1: See the lecture notes of Week 6. (b) Problem 2: See the lecture notes of Week 7 and the supplementary exercises of Week 7. 23 stv A chool Thomam 1. (Subspace Criteria): (will be nod to gium Bulevat forma . slapen or not of the onbent v.s.) whether vector ower RR V be det w be a subset of v. W is a subspace of v it and only if (1) Winonempty. (:) utvEw whereves new and vEW. () duEw NEW LGIR. Proof! Now it wise ( solapara by definition w tall sa WS. with the operations inherited from V. Then closely (i), (i), (..) hobs. 1 that (i), (i), (ii) holls. By previous remand we nord to verify: 0,6 W. Since Wis nonempty by (i), IwEW. consider OvEw by v. by fact (1) For any UEW,-uEW. (-1) a=-"EW by v by foct (1) Then OFIR Nas console