Question

I need help with these linear algebra problems.

1. Consider the following subsets of R³. Explain why each is is not a subspace.

(a) The points in the xy-plane in the first quadrant.

(b) All integer solutions to the equation x² + y² = z² .

(c) All points on the line x + z = 5.

(d) All vectors where the three coordinates are the same in absolute value.

2. In each of the following, state whether it is a vector space. Justify your answer.

(a) the set of all polynomials with degree exactly 1

(b) the set of all 2 × 2 matrices with determinant 2

(c) the set of all diagonal 3 × 3 matrices

(d) the set of all vectors in R⁴ whose entries sum to 0.

(e) the set of all antiderivatives of f(x) = x⁵

Answer 1

1. (a) w = }(X,Y, 2) ER: x20 YZ0,2=0} is not a subspace of Rs as (1,1,0) & W but its additive inverse (+1,-1,0) & ut. + (b) W ={(x, y, 2) = 2: x + y = 7² } is not a subspace of IR? since (3,4,5) EW and I ER but (3,4,5) = 62,3,5) $TT. (6) w = {(x, y, z) ER: x+2=5} is not a subsface of Re since (0,0,0) the additive identity of 1 is not in Wind (d) W = {(x, y, z) ElR?: 121=171-121) is not a subspace of Rs since (1,1,1) EW and (-1,1,DEW but (1,1,1)+(-1,1,1)= (0,2,2) & W 2. (a) NOT a vector space as 1+x and 1-x both are in the set but (1+x) + (1-x)=2 has degree o so a doesn't belong to the set (other reason o & set) (6) NOT a Vector space Note (20) and (62) Note 2 o ) and / of both are in the set (20) + (60) = (3 ) has determinan q, which is not in the set (other reason (ools & set ) (e) Bi Yes, it is a vector space. (ooo being an element it is non-empty set. Clearly it is closed under addition and scalar multiplication.

(d) Yes, it is a vector space. Since (6,0,0,0) is in the set it is non-empty. Kleasly it is closed under so addition and scala multiplication. - (C) No imbe ) Since (6)=0&x bene o is not Huan antidesivative of X5 M dx