Understanding Basis for P2

The video tutorial explains how to determine if a set of polynomials forms a basis for the polynomial space P2. It discusses two approaches: checking for independence and checking if the set spans P2. The tutorial demonstrates converting polynomials to vectors and forming augmented matrices to test for independence. It evaluates three sets of polynomials, concluding that the second and third sets form a basis for P2, while the first set does not.

What is the minimum number of polynomials required in a set to form a basis for P2?

• Five

• Two

• Three

• Four

Which method is used in the video to determine if a set of polynomials forms a basis for P2?

• Checking if the set is complete

• Checking if the set is orthogonal

• Checking if the set is independent

• Checking if the set spans P2

What does it mean if a set of vectors is dependent?

• The vectors are orthogonal

• The vectors are linearly independent

• The vectors do not form a basis for R3

• The vectors form a basis for R3

In the context of the video, what is a 'free variable'?

• A variable that is dependent on others

• A variable that is constant

• A variable that can take any value

• A variable that is always zero

What is the result when the first set of polynomials is tested for independence?

• The set is dependent and does not form a basis

• The set spans P2

• The set is independent and forms a basis

• The set is orthogonal

How is the second set of polynomials verified to form a basis for P2?

• By checking if it spans P2

• By checking if it is independent

• By checking if it is orthogonal

• By checking if it is complete

What indicates that the second set of polynomials forms a basis for P2?

• The set spans R3

• The set is orthogonal

• The set is dependent

• The set is incomplete

What is the outcome of testing the third set of polynomials?

• The set does not span P2

• The set forms a basis for P2

• The set is dependent

• The set is incomplete

What is the significance of having three pivots in the augmented matrix for the third set?

• It indicates the set is dependent

• It indicates the set is incomplete

• It indicates the set is independent

• It indicates the set is orthogonal

What conclusion is drawn about the third set of polynomials?

• It is orthogonal

• It does not span P2

• It is independent and forms a basis

• It is dependent and does not form a basis