# Basis of a vector space

## 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?

## 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.

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## 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?

## 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?

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?