Is the set of all polynomials of P3 such that p 2 0 a subspace of P3 Why or why not?

Yes! Since every polynomial of degree up to 2 is also a polynomial of degree up to 3, P2 is a subset of P3.

What is the zero vector for polynomial?

The zero vector is given by the zero polynomial. The degree of the polynomials could be restricted or unrestricted. For example, one could consider the vector space of polynomials in x with degree at most 2 over the real numbers, which will be denoted by P2 from now on.

How do you prove that a vector is zero?

Proof (a) Suppose that 0 and 0 are both zero vectors in V . Then x + 0 = x and x + 0 = x, for all x ∈ V . Therefore, 0 = 0 + 0, as 0 is a zero vector, = 0 + 0 , by commutativity, = 0, as 0 is a zero vector. Hence, 0 = 0 , showing that the zero vector is unique.

How do you find the basis of a polynomial vector space?

A basis for a polynomial vector space P={p1,p2,…,pn} is a set of vectors (polynomials in this case) that spans the space, and is linearly independent.

How do you prove a vector is zero?

What is the value of p(0) = p(1) = 0?

Specifying p(0) = p(1) = 0 means that any polynomial in W must be divisible by x and (x − 1). That is W = {x(1 − x)p(x) | p(x) ∈ P1}. Since P1 has dimension 2, W must have dimension 2. Extending W to a basis for V just requires picking any two other polynomials of degree 3 which are linearly independent from the others.

How do you know if a p-value is statistically significant?

The smaller the p-value, the stronger the evidence that you should reject the null hypothesis. A p-value less than 0.05 (typically ≤ 0.05) is statistically significant. It indicates strong evidence against the null hypothesis, as there is less than a 5\% probability the null is correct (and the results are random).

What does a p-value of 0 mean in a research study?

The p -value is conditional upon the null hypothesis being true is unrelated to the truth or falsity of the research hypothesis. A p -value higher than 0.05 (> 0.05) is not statistically significant and indicates strong evidence for the null hypothesis. This means we retain the null hypothesis and reject the alternative hypothesis.

What is the p-value of the null hypothesis?

The p-value is conditional upon the null hypothesis being true, but is unrelated to the truth or falsity of the alternative hypothesis. A p-value higher than 0.05 (> 0.05) is not statistically significant and indicates strong evidence for the null hypothesis. This means we retain the null hypothesis and reject the alternative hypothesis.