Understanding Basis for P3

The video tutorial explains the concept of the dimension of P3, the set of polynomials of degree three or less, and how to determine a basis for it. It discusses the elimination of sets that cannot form a basis due to the number of polynomials or lack of independence. The tutorial demonstrates the process of testing polynomial sets for independence using vector equations and augmented matrices. It concludes by confirming which sets form a basis for P3, emphasizing the importance of having four independent polynomials to span the space.

What is the dimension of the set of polynomials P3?

• 5

• 4

• 3

• 2

Why can't a set with five polynomials be a basis for P3?

• It includes a zero polynomial.

• It has too few polynomials.

• It doesn't include a constant term.

• It has too many polynomials.

What is the first step in checking if a set of polynomials is independent?

• Count the number of polynomials.

• Solve the vector equation.

• Check for zero polynomials.

• Convert polynomials to vectors.

What does it mean if a set of vectors only has the trivial solution?

• The set is dependent.

• The set has infinite solutions.

• The set is independent.

• The set cannot form a basis.

How many pivots should a set of vectors have to be considered independent?

• One pivot

• Two pivots

• Three pivots

• Four pivots

What indicates that a set of vectors is dependent?

• It has a row of zeros.

• It has a pivot in every column.

• It has only the trivial solution.

• It spans R4.

Which of the following sets can form a basis for P3?

• A set with three polynomials

• A set with five polynomials

• A set with a zero polynomial

• A set with four independent polynomials

What is the role of the constant term in determining a basis for P3?

• It can be any value.

• It must be non-zero.

• It must be zero.

• It is irrelevant.

Why is a set with a free variable not a basis for P3?

• It has infinite solutions.

• It has no solutions.

• It has only the trivial solution.

• It spans R3.

How many sets were identified as valid bases for P3 in the video?

• One set

• Four sets

• Two sets

• Three sets