Hello Im in dire need of solving these Linear Algebra questi

Hello, I\'m in dire need of solving these Linear Algebra questions since I do not know how to do it. Please explain the clearest and easiest way possible!

1). Let S be the subset of P4 consisting of all polynomials p(x) of the form p(x) = ax^3+bx^2+cx+2a+b+3c.

a. Show that S is a subspace of P4

b. Find a basis and the dimension of S

Solution

Let S be the subset of P4 consisting of all polynomials p(x) of the form p(x) = ax^3+bx^2+cx+2a+b+3c.

Part(a) We have to show that S is a subspace of P4

Sol:-

(i) first we have to check S is empty or not ?

Consider a example x^3+x^2+x+6 which is belong to S

so S is not empty

(i) Next we have to check if S is closed under addition

let P1 and P2 are two polynomial in S

P1= ax^3+bx^2+cx+2a+b+3c

P2 = ux^3+vx^2+wx+2u+v+3w

(P1+P2)(x) = (a+u)x^3 + (b+v)x^2 + (c+w)x + 2a+b+3c+2u+v+3w

(P1+P2)(x)   is also belong to S

(iii) Next we check if S is closed under scalar multiplication

let t is a scaler number

then (tP)(x) = (ta)x^3+(tb)x^2+(tc)x+(2a+b+3c)t

is also belong to S

So S is a subspace of P4

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part(b) Find a basis and the dimension of S

basis - {x, x^2, x^3}

dimension = 3

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Ask if any doubt !!!!