Show that the following polynomials form a basis for P3 1+x,...

Key Concepts

Polynomial Vector Space

A polynomial vector space, such as P?, consists of all polynomials of degree at most 3. This space is equipped with operations of polynomial addition and scalar multiplication, which satisfy the conditions required of a vector space, making it a fundamental example in linear algebra.

Basis

A basis of a vector space is a set of vectors that is both linearly independent and spans the entire space. In the context of polynomial spaces, a basis allows any polynomial in the space to be uniquely written as a linear combination of the basis polynomials.

Linear Independence

Linear independence of a set of vectors (or polynomials) means that no non-trivial linear combination of these vectors yields the zero vector. For the given set of polynomials, demonstrating linear independence ensures that none of the polynomials can be written as a combination of the others.

Span

To span a vector space means that any vector (polynomial in this case) within the space can be expressed as a linear combination of the vectors in the set. Showing that the set spans the space P? confirms that every polynomial of degree at most 3 can be formed from the given polynomials.

Dimension

The dimension of a vector space is the number of vectors in any basis for the space. Since P? is a 4-dimensional space, any set of four linearly independent polynomials that spans P? will form a basis for it.

Transcript

00:01 Hi guys.

00:02 My name is julian, and i'll be helping you solve today's linear algebra problem.

00:06 So we have these four polynomials.

00:08 We want to show that these four polynomials are a basis for p3, the vector space of all polynomials up to a degree three.

00:15 Our requirements for a basis are that a set of vectors spans their vector space and that they are linearly independent.

00:22 So for them to span a vector space, that means that any vector in our vector space v, right? so any v in v can be represented as a linear combination of our basis vectors, or hypothetical basis vectors in this case, and that our hypothetical basis factors are linearly independent, so they cannot be written as a linear combination of each other.

00:46 So as long as we can check off both these boxes, that means that our set of vectors is indeed a basis.

00:52 So let's try and check off the spanning requirement first.

00:56 So first we grab an arbitrary polynomial from p3, right? and any polynomial in p3 will look just like this.

01:17 And so now we want to set up an equation.

01:20 So we want to set up a linear combination of these boys to equal our right -hand side here.

01:27 So we'll just grab some arbitrary constants.

01:31 I'll call them b -0 -1 plus b -1 -x plus b -1 -1 -1 -1 -1 -x.

01:39 Plus b2 1 minus x squared plus b3 1 minus x cubed and we'll set that equal to p3 of x of x and so let's expand this out and then collect our terms by degree so we get b0 plus b0 x plus b1 minus b1 x plus b2 minus b1 x plus b2 minus b2x squared plus b3 minus b3 x cubed, and this should equal are arbitrarily selected p3.

02:29 And mind you, these are arbitrary constants, so these ai could be anything.

02:34 And that's precisely what makes it an arbitrary vector.

02:39 And so let's collect our terms by degree now.

02:43 So we only have one term of degree three, and that's negative b3.

02:47 And then we have one term of degree two.

02:56 And then we have one, two terms of degree one.

03:04 So we have b0 minus b1 times x...

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