1 point) Let V-Ps, the vector space of all polynomials of degree 5 or less. (a)
ID: 3138162 • Letter: 1
Question
1 point) Let V-Ps, the vector space of all polynomials of degree 5 or less. (a) dim V 6 (b) If U is the subspace of V consisting of all polynomials of degree less than 5, then dim U - (c) If U is the subspace of V consisting of all polynomials of degree 3 or less, then dim U- (d) If U is the subspace of V consisting of all polynomials with constant term equal to 0, then dim U - (e) If U is the subspace of V consisting of all polynomials involving only odd powers of x, then dim U- (f) If U is the subspace of V consisting of all polynomials involving only even powers of x (including the constant term), then dim U (g) If U is the subspace of V consisting of all polynomials whose coefficients sum to 0, then dimU -Explanation / Answer
True as {1,x,x2,x3,x4,x5} is the standard basis for P5. {1,x,x2,x3,x4} is the standard basis for U so that dim(U) = 5. {1,x,x2,x3} is the standard basis for U so that dim(U) = 4. dim(U) = 5 as {x,x2,x3,x4,x5} is the standard basis for U. dim(U) = 3 as {x,x3,x5} is the standard basis for U. ( 0 is considered to be even). dim(U) = 3 as {1,x2,x4} is the standard basis for U. a+bx+cx2+dx3+ex4+fx5 is an arbitrary vector in P5. If a+b+c+d+e+f = 0, then a = -(b+c+d+e+f) so that a+bx+cx2+dx3+ex4+fx5 = -(b+c+d+e+f)+ bx+cx2+dx3+ex4+fx5 = b(x-1)+c(x2-1)+d(x3-1)+e(x4-1) +f(x5-1). Hence {(x-1),(x2-1),(x3-1),(x4-1),(x5-1)}is a basis for U so that dim(U) =5.
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