What is a Basis of a Vector Space and How to Find Another Basis?

[Coolster7]

Homework Statement

There is a standard basis, B = (1; z; z^2; z^3; z^4) where B is the basis of a R4[z] of real polynomials of at most degree 4.

I need to find another basis B' for R4[z] such that no scalar multiple of an
element in B appears as a basis vector in B' and also prove that this B' is a basis.

Can any help with this please?

Homework Equations

The Attempt at a Solution

I could only think to do a basis B'=(0,1,z,z^2,z^3) but no sure if this is correct.

 

[Mark44]

Homework Statement

There is a standard basis, B = (1; z; z^2; z^3; z^4) where B is the basis of a R4[z] of real polynomials of at most degree 4.

I need to find another basis B' for R4[z] such that no scalar multiple of an
element in B appears as a basis vector in B' and also prove that this B' is a basis.

Can any help with this please?

Homework Equations

The Attempt at a Solution

I could only think to do a basis B'=(0,1,z,z^2,z^3) but no sure if this is correct.

No, it isn't correct. The 0 vector can never be a vector in a basis for a vector space.

Here's a hint: the function 1 + z is not a linear multiple of any of the vectors in B, right?

 

[Coolster7]

No, it isn't correct. The 0 vector can never be a vector in a basis for a vector space.

Here's a hint: the function 1 + z is not a linear multiple of any of the vectors in B, right?

So you could have B' = (1+z, 1+z^2, 1+z^3, 1+z^4, 1+z^5) for example or would this not work because the highest degree for R4[z] is 4?

Thanks for your reply.

 

[Mark44]

Right, you can't have 1 + z⁵. Can't you think of any other combinations besides adding 1 to another vector in the basis?