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

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A set of 6 vectors in R5 cannot be a basis for R5 because it is linearly dependent. A basis must consist of linearly independent vectors that span the space, and since the dimension of R5 is 5, any set of vectors exceeding this dimension will inherently be dependent. Therefore, the statement is true, confirming that a basis cannot have more vectors than the dimension of the space it spans.

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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?
 
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NewtonianAlch said:

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.
 
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.
 
Thanks for the help.
 

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