Determine whether the following are subspaces of P 4 (Be Careful!) (P 4 is defin
ID: 2939971 • Letter: D
Question
Determine whether the following are subspaces of P4 (Be Careful!)(P4 is defined as the set of all polynomialsof degree less than four)
The set of all polynomials of degree 3
and
The set of all polynomials p(x) in P4 suchthat p(0) = 0
This one is giving me quite a bit of trouble! I want to beable to explain this and work it out, rather than be able to seethat the first is not a subspace and the second is a subspace.Thank you very much.
(P4 is defined as the set of all polynomialsof degree less than four)
The set of all polynomials of degree 3
and
The set of all polynomials p(x) in P4 suchthat p(0) = 0
This one is giving me quite a bit of trouble! I want to beable to explain this and work it out, rather than be able to seethat the first is not a subspace and the second is a subspace.Thank you very much.
The set of all polynomials of degree 3
and
The set of all polynomials p(x) in P4 suchthat p(0) = 0
This one is giving me quite a bit of trouble! I want to beable to explain this and work it out, rather than be able to seethat the first is not a subspace and the second is a subspace.Thank you very much.
Explanation / Answer
The set of all polynomials of degree 3 is not a subspace of P4 because the first and second conditions of a subspace are not satisfied. To see this consider the following elements of the set given in (b): x3 + x2 + x and ?x3 + x2 + x. We can see the first condition is not satisfied because if we take the scalar zero and the first element given above we obtain: 0(x3 + x2 + x) = 0. Clearly zero is not a polynomial of degree 3 and the first condition of a subspace is not satisfied. When the two elements above are added together we obtain: (x3 + x2 + x) + (?x3 + x2 + x) = 2x2 + 2x. Clearly 2x2 + 2x is not a polynomial of degree 3 and the second condition of a subspace is not satisfied.
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