Question

Determine whether or not each of the following set is a subspace of P₃(R). if it is a subspace, then find a basis and its dimension.

(a) U = p(1) =0

Answer 1

The set U = p(1) = 0 is a subspace of P₃(R) with a dimension of 2. A possible basis for U is {1 - x, x² - 1}.

Step-by-step explanation:

To determine whether the set U = p(x) in P₃(R) is a subspace of P₃(R), we need to check if it satisfies three conditions:

1. The zero vector is in U.
2. If u and v are in U, then u + v is also in U.
3. If u is in U and c is a scalar, then cu is also in U.

Let's check each condition:

1. The zero vector, which is p(x) = 0, evaluates to 0 at x = 1. Therefore, the zero vector is in U.
2. If p₁(1) = 0 and p₂(1) = 0, then (p₁ + p₂)(1) = p₁(1) + p₂(1) = 0 + 0 = 0. Therefore, the sum of two polynomials in U will also evaluate to 0 at x = 1, and thus, it is in U.
3. If p(1) = 0, then (cp)(1) = c(p(1)) = c(0) = 0. Therefore, multiplying a polynomial in U by a scalar will also result in a polynomial that evaluates to 0 at x = 1, and thus, it is in U.

Since all three conditions are met, U is a subspace of P₃(R). Now, let's find the basis and dimension of U:

Since p(1) = 0, we can write an arbitrary polynomial p(x) as p(x) = a₀ + a₁x + a₂x². Substituting x = 1, we get a₀ + a₁ + a₂ = 0. This equation relates a₀, a₁, and a₂, and it represents a plane in a three-dimensional space. Therefore, the dimension of U is 2, and a possible basis for U is {1 - x, x² - 1}.