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}
To determine if a subset is a subspace of a vector space, we need to check if it satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector.
Let's check the given subsets:
a) {p(t)|the constant term of p(t) is 1}
To check if this subset is a subspace, we need to see if it satisfies all three conditions.
i) Closure under addition:
Take two polynomials p(t) and q(t) from the subset, where p(t) has a constant term of 1 and q(t) has a constant term of 1. When we add these two polynomials, we obtain a polynomial with a constant term of 2 (1 + 1). Therefore, the subset is not closed under addition.
Since it fails the closure under addition test, this subset is not a subspace.
e) {p(t)|the degree of p(t) is 3}
For this subset, we need to again check if it satisfies all three conditions.
i) Closure under addition:
Take two polynomials p(t) and q(t) from the subset, where p(t) has a degree of 3 and q(t) has a degree of 3. When we add these two polynomials, we obtain a polynomial with a degree of 3 (if we consider the highest power term). Therefore, the subset is closed under addition.
ii) Closure under scalar multiplication:
Take a polynomial p(t) from the subset, where p(t) has a degree of 3 and multiply it by a scalar c. The resulting polynomial, cp(t), will also have a degree of 3 (as multiplying each term by a scalar does not change the degree). Therefore, the subset is closed under scalar multiplication.
iii) Contains the zero vector:
The zero vector in R5[t] is the polynomial with all coefficients equal to zero. In this case, the zero vector has a degree of 0. Since a degree of 0 satisfies the condition of having a degree of 3, the subset contains the zero vector.
Since this subset satisfies all three conditions, it is a subspace.
In summary, only the subset e) {p(t)|the degree of p(t) is 3} is a subspace of R5[t].