A set of 6 vectors in R5 cannot be a basis for R5, true or false?

[NewtonianAlch]

Homework Statement

A set of 6 vectors in R5 cannot be a basis for R5, true or false?

The Attempt at a Solution

I'm thinking true, because any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5.

To be a basis it must be a linearly independent spanning set, so if it's linearly dependent, it cannot be a basis.
Am I correct?

 

[Mark44]

Homework Statement

A set of 6 vectors in R5 cannot be a basis for R5, true or false?

The Attempt at a Solution

I'm thinking true, because any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5.

To be a basis it must be a linearly independent spanning set, so if it's linearly dependent, it cannot be a basis.
Am I correct?

Yes.

 

[HallsofIvy]

A "basis" for a finite dimensional vector space has three properties:
1. It spans the space.
2. Its vectors are independent.
3. The number of vectors in the basis is equal to the dimension of the space.

And, if any two of these are true, so is the third.

 

[NewtonianAlch]

Thanks for the help.