Let P_3 be the vector space of polynomials of degree less than 3. Demonstrate th

Question

Let P_3 be the vector space of polynomials of degree less than 3. Demonstrate that {x^2 + x, x^2 -x, 2x] is not a spanning set for P_3 by finding a polynomial in P_3 not in its span: You can probably solve this one easily by guessing, but try different answers until you figure out a criterion for not being in the span. What is this criterion? Demonstrate that {x^2 + 2x- 3, x + 1, x^2 + 4x - 1} is not a spanning set for P_3 by finding a polynomial in P_3 not in its span: On the other hand, [-1, x - 2, x^2 + x} is a spanning set for P_3. Illustrate this by writing each of the following polynomials as a linear combination of members of this set (a) x =, (-1) +, (x - 2) +, (x^2 + x) (b) x^2 + 3x + 1 =, (-1) +, (x - 2) +, (x^2 + x)

Explanation / Answer

x is polynomial in P3(R) but it cannot be written as a linear compination of x^2+x,x^-x,2x.

P3 is a vectorspace of dimension 3 so instead of this we can check the given set is linearly independent.if it is linearly independent we can say that 3 dimensional linearly independent set in 3 dimensional vectorspace also generates P3 so its spans P3.

the polynomial x^2+x+1 cannot be written as a linear combination of {x^2+2x-3,x+1,x^+4x-1}.

x=(-2)(-1)+1*(x-2)+0(x^2+x)

x^2+3x+1=(-5)(-1)+2(x-2)+1*(x^2+x)

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