Question

linear algebra

2. Which of the following subsets of Rare actually subspaces? Justify your answer in terms of the definition and properties of subspaces. (a) The vectors [x y z]" with x + 2y -z = 0. (b) The vectors [a b c]" with a + b + c = 3. (c) The vectors [a+2bb-3b]' where a, b are any real numbers, (d) The vectors [pr] where q.r are any real numbers and p20.

Answer 1

Solution Do 22 @ the rectors () with x+2y-z=0 → S, (let) here (0,0,0) E R and (0,0,0)T ES, & S, is non-empty. Now, let (^,, y, z) ES, and $ (N2, Yz, 22) E S₂ . x+2y, -z=0 so, X₂ +24₂ - 2q =0 So, 2, +24,-2, + a2 +2 4 ₂ - 2 = (2,+ 12) + (@y,+2yz) -(2+2) = 0 » vector addition satisfier . Also, let (^1,9, 2) ES, & + 2y, - 2,20 so, d(a,+ 2y, - 2) = Lai + 2xy, - az, zo - Scalar multiplication satisfies (Hence S, is subspace) a+267 Ⓡ$ - [67 with at btc = 3, Now (0,0,0)T & Sse is A not a rectar space b li a, b, c are real nos 1 -36 J 7 for any a b c ER, (0,0,0) e S3 Non empty Now, for any real nos rector addition and scalar multiplication

satisfies z is a subspace a q), per q,0 ER, PZO so, p is any koritise number, Now, if we take per the scalar multiplication property say he multiply by ra, then we here -ala. Fap 1 I then - aq ER, - an GIR hence but rap is not donot satisfies scalar o Multiplication Not a subspace