Vector Spaces: A Comparison of Two Bases in V3(R)

[ashnicholls]

Are these two sets:

A = {(0,2,2)^T, (1,0,1)^T, (1,2,1)^T}

B= {(1,2,0)^T, (2,0,1)^T, (2,2,0)^T}

Bases of V3(R)

I have found equations that show that they span V3(R)

And that both set are linearly independant.

So am I right in saying that they are both bases of V3(R).

Cheers Ash

 

[matt grime]

If you have three vectors in R^3 that span, then they must be linearly independent, and if you have three linearly independent vectors in R^3 then they must span. That follows from the definition of span and basis.

 

[tacman]

Yes, you are correct in saying that both sets A and B are bases of V3(R). In order for a set to be considered a basis of a vector space, it must satisfy two conditions: linear independence and spanning the space.

From the information provided, we can see that both sets A and B are linearly independent, meaning that none of the vectors in each set can be written as a linear combination of the other vectors. This is an important property for a basis because it ensures that there is no redundancy in the set and each vector is necessary to span the entire space.

Additionally, both sets A and B span V3(R), meaning that any vector in V3(R) can be written as a linear combination of the vectors in the set. This is also a crucial property for a basis because it ensures that the set can generate all possible vectors in the space.

Therefore, based on the information provided, we can conclude that both sets A and B are indeed bases of V3(R).