I\'m confused about how to set-up from section 4.1 question 7 and Q8, but if I c

Question

I'm confused about how to set-up from section 4.1 question 7 and Q8, but if I correctly understand Q7 then Q8 should be easier to answer, which Q7 states: 7. All polynomials of degree at most 3, with integers as coefficients. The key is to determine if the given set is a subspace of Pn. Given A =2 B=3 t = 2 P(t) =A0+A1t+A2t^2+A3t^3=2+4+8+16=30 Q(t)=B0+B1t+B2t^2+B3t^3=3+6+12+24=45 To be a subspace, three requirements must be satisfied: 1. The subspace must contain the zero vector within Pn 2. The subspace must be closed under addition for Pn 3. The subspace must be closed under scalar multiplication for Pn Zero vector: P(0) =A0t^0 where the power is zero where c=0; A = -5; c*P(t)=c*P(t)=0*P(3)= 0*(-5)=0 Addition of polynomials: Given A =2 B=3 t = 2 P+Q =(A0+A1t+A2t^2+A3t^3)+(B0+B1t+B2t^2+B3t^3) P+Q =(A0+A1t+A2t^2+A3t^3)+(B0+B1t+B2t^2+B3t^3)=(A0+B0)+(A1+B1)t+(A2 +B2)t^2+(A3 +B3)t^3=5+5*2+5*2^2+5*2^3=15+20+40=75 I see the sum of P+Q is 75, since the individul integers of P(t) and Q(t) when added together provide the same answer. Scalar multiplication of polynomials: I know the end point of C*A3t^3 is an odd function, and polynomials are continuous functions, but this does not explain whether Pn is a subspace. where C= (-5) A=2, t=3 (C*P)(t)=C*P(t) =C*A0+C*A1t+C*A2t^2+C*A3t^3= -10-30-90-270=-400 it's also obvious to design the above scalar to have a positive value. Since the author of Q7 said I could use integers, then all integers must be in Pn (the zero vector exist, the set is closed under scalar multiplication and closed under addition). So the big question is, does the above Q7 with the detailed answer(s) look correct from a stand point of have I learned anything from section 4.1? //signed// Mike Simpson

Explanation / Answer

Not sure what you want here, but I'll give it a shot.
In the following, we will consider following subsets of

A set is "closed under addition" if the sum of any two members of the set also belongs to the set. For example, the set of even integers. Take any two even integers and add them together. The result is an even integer.

A set is "closed under (scalar) multiplication" if the product of any member and a scalar is also in the set. In other words, if x is in S and a is any scalar then ax will be in the set if the set is closed under scalar multiplication. For example, the set of 2 x 2 diagonal matrices is closed under scalar multiplication.

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