Let R5[t] be the vector space of all polynomials in t of degree 4 or less with real coefficients. Which of the following subsets are subspaces?
I know how to test if something is a subspaces, just not sure how to do it with for these polynomial equations.
a.) {p(t)|the constant term of p(t) is 1}
e.) {p(t)|the degree of p(t) is 3}
a) If you take two polynomials p1(t) and p2(t) that have a constant term 1 and take the linear combination
a p1(t) + b p2(t)
then what is the constant term of that linear combination? For arbitrary a and b does this then belong to the set of all the 4th degree polynomials that have a constant term of 1?
To determine whether a subset is a subspace, we need to verify three conditions: closure under addition, closure under scalar multiplication, and the presence of the zero vector.
a.) {p(t) | the constant term of p(t) is 1}
To check if this subset is a subspace, we need to ensure that all polynomials in this subset satisfy the three conditions.
1. Closure under addition: Suppose we have two polynomials p(t) and q(t) in the subset. We need to show that their sum, p(t) + q(t), also belongs to the subset. Since the constant term of p(t) is 1 and the constant term of q(t) is 1, the constant term of their sum will also be 1. Therefore, this subset satisfies closure under addition.
2. Closure under scalar multiplication: Let p(t) be a polynomial in the subset and c be a scalar. We need to show that c * p(t) is also in the subset. Since the constant term of p(t) is 1, multiplying it by any scalar c will still result in a polynomial with a constant term of 1. Hence, this subset satisfies closure under scalar multiplication.
3. Presence of the zero vector: The zero vector in this vector space is the polynomial 0(t), which has all coefficients equal to zero. However, the constant term of 0(t) is 0, not 1. Therefore, this subset does not contain the zero vector.
Since this subset fails to satisfy the third condition, it is not a subspace of R5[t].
e.){p(t) | the degree of p(t) is 3}
Again, we need to verify the three conditions for this subset.
1. Closure under addition: Suppose we have two polynomials p(t) and q(t) in the subset. We need to show that their sum, p(t) + q(t), also belongs to the subset. If the degree of p(t) is 3 and the degree of q(t) is also 3, then the sum, p(t) + q(t), will have a degree of 3 as well. Therefore, this subset satisfies closure under addition.
2. Closure under scalar multiplication: Let p(t) be a polynomial in the subset, and c be a scalar. We need to show that c * p(t) is also in the subset. If the degree of p(t) is 3, multiplying it by any scalar c will still result in a polynomial with a degree of 3. Hence, this subset satisfies closure under scalar multiplication.
3. Presence of the zero vector: The zero vector in this vector space is the polynomial 0(t), which has all coefficients equal to zero. A polynomial of degree 3 with all coefficients equal to zero is indeed the zero vector. Therefore, this subset contains the zero vector.
Since this subset satisfies all three conditions, it is a subspace of R5[t].