Determine if each of the following sets is a subspace of ℙn, for an appropriate value of n. Type "yes" or "no" for each answer.

Let W1 be the set of all polynomials of the form p(t)=at2, where a is in ℝ.
Let W2 be the set of all polynomials of the form p(t)=t2+a, where a is in ℝ.
Let W3 be the set of all polynomials of the form p(t)=at2+at, where a is in ℝ.

1. Yes.

2. No.

3. Yes.

Step-by-step explanation:

Consider the following subsets of Pn given by

1.Let W1 be the set of all polynomials of the form , where a is in ℝ.

2.Let W2 be the set of all polynomials of the form , where a is in ℝ.

3. Let W3 be the set of all polynomials of the form , where a is in ℝ.

Recall that given a vector space V, a subset W of V is a subspace if the following criteria hold:

- The 0 vector of V is in W.

- Given v,w in W then v+w is in W.

- Given v in W and a a real number, then av is in W.

So, for us to check if the three subsets are a subset of Pn, we must check the three criteria.

- First property:

Note that for W2, for any value of a, the polynomial we get is not the zero polynomial. Hence the first criteria is not met. Then, W2 is not a subspace of Pn.

For W1 and W3, note that if a= 0, then we have p(t) =0, so the zero polynomial is in W1 and W3.

- Second property:

W1. Consider two elements in W1, say, consider a,b different non-zero real numbers and consider the polynomials

.

We must check that p_1+p_2(t) is in W1.

Note that

Since a+b is another real number, we have that p1(t)+p2(t) is in W1.

W3. Consider two elements in W3. Say . Then

So, again, p1(t)+p2(t) is in W3.

- Third property.

W1. Consider an element in W1 and a real scalar b. Then

.

Since (ba) is another real scalar, we have that bp(t) is in W1.

W3. Consider an element in W3 and a real scalar b. Then

.

Since (ba) is another real scalar, we have that bp(t) is in W3.

After all,

W1 and W3 are subspaces of Pn for n= 2

and W2 is not a subspace of Pn.  

in mathematics, a function is a relationship between two variables such that the value of one of the variables, called the dependent variable, is determined by the value of the other variable, called the independent variable. there is a specific term we use to describe the set of independent variables of a function.

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