a A theorem from class says that if we have 5 vectors in P4

a) A theorem from class says that if we have 5 vectors in P4, then they have to be linearly dependent. But this theorem says nothing about whether or not these vectors span the space. Give (1) an example of 5 vectors in P4 which span the space, and (2) an example of 5 vectors in P4 which do not span the space.

b) Another theorem from class implies that if we have 3 vectors in P4, then they do not span the space. But this theorem says nothing about whether these vectors are linearly independent. Give (1) an example of 3 vectors in P4 which are linearly independent, and (2) an example of 3 vectors in P4 which are linearly dependent.

Note: In this problem, you do not need to prove that your examples have the desired properties

Solution

(a) The basis in P⁴ has four independent vectors. Out of these 5 vectors, we can have a maximum of 4 vectors which are lineary independent, Therefore, the fifth vector will be linearly dependent and can be expressed as a linear combination of (some or all) the remaining vectors.

Example 1. The five vectors in P⁴ (the vector space of all polynomials of degree. < 4) which span the space

   x³ + 1, x² - x + 2, x², x, x + 1.

  The five vectors in P⁴ (the vector space of all polynomials of degree. < 4) which do not span the space:

x³ + 2x, x³ - x² + 2, 3x³+3x, 2x, 5x.

(b) Since, P⁴ is the vector space of all polynomials of degree. < 4. Therefore,it can have a subset with a maximum four independent vectors. Remaining vectors will be linearly dependent.

Example 2. The three vectors in P⁴ (the vector space of all polynomials of degree. <4) which are linearly independent vectors

     x³ + 1, x³ - x² + 2, x²+ 2x + 1.

The five vectors in P⁴ (the vector space of all polynomials of degree. 4) which do not span the space:

x³ + 2x + 2, 2x³ + 4x + 4, 3x³ + x + 2.