Let 𝒫^(4) denote the vector space consisting of all...

Let $\mathcal{P}^{(4)}$ denote the vector space consisting of all polynomials $p(x)$ of degree $\leq 4$.
(a) Are $x^3-3 x+1, x^4-6 x+3, x^4-2 x^3+1$ linearly independent elements of $\mathcal{P}^{(4)}$ ?
(b) What is the dimension of the subspace of $\mathcal{P}^{(4)}$ they span?

Key Concepts

Span

The span of a set of vectors is the collection of all possible linear combinations of these vectors. In the context of a polynomial vector space, the span of a set of polynomials is a subspace that includes every polynomial that can be expressed using the given set as a basis for that subspace.

Dimension of a Subspace

The dimension of a subspace is defined as the number of vectors in a basis for that subspace. It indicates the minimum number of independent directions required to represent any element within the subspace. In polynomial spaces, this corresponds to the number of independent polynomial terms needed to form any polynomial in the subspace.

Polynomial Vector Space

A polynomial vector space is a set of polynomials that forms a vector space under the usual operations of addition and scalar multiplication. In this context, the vector space is comprised of all polynomials of degree at most a certain number, where each polynomial can be uniquely represented by a linear combination of basis polynomials associated with each degree up to that maximum.

Linear Independence

Linear independence is a property of a set of vectors where no nontrivial linear combination of these vectors results in the zero vector. For polynomials, this means that the only way to combine them to yield the zero polynomial is by having all coefficients in the combination equal to zero.

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