Question

When can a determinant be used to figure out whether a set of vectors is a basis? For example, the coefficient matrix can be used to determine whether a set is linearly independent and whether it spans, but obviously there are circumstances in which a set is linearly independent, but does not span and vise versa.

Answer 1

I know that is the determinant of 3 vectors is 0 it means that they are not a basis for R3

Answer 2

So the rule must be that one can determine whether a set is a basis for a vector space if the set has the same dimension? For example, a set of two vectors that are linearly independent does not span R3. So, in this case, you cannot simply take the determinant of the coefficient matrix because it may vary well tell you that the determinant is 0, but really it is not a basis at all because it does not span the vector space.

Answer 3

Theorem:
A Set of vectors is linearly independent if the determinate of the set of vectors is not equal to zero.

If you are looking for a set of vectors to be a basis,
1) Check to see that your dimensions are correct, if your dimension is 5 for example, you will require 5 vectors that are linearly dependent.
2) take the det of those vectors and make sure they are not zero.

then the RREF of the set of vectors will be a basis e.

Answer 4

agree/disagree?

Answer 5

i'm not sure what you mean by 1) but I agree with 2). What do you mean i will "require"?

Answer 6

if you are worried about Span then you will need the same number of vectors as you have for dimensions.

Answer 7

What would a be requiring 5 vectors to be linearly dependent for? Don't the vectors have to be linearly independent to be a basis for a vector space?

Answer 8

for example, If you have a 5 dimensional space, you will need 5 vectors that are linearly independent to serve as a basis. 4 vectors will not span all of 5 dimensions, more than 5 vectors will be linearly dependent.

Answer 9

Okay!...that's what I thought! So, if I have the same number of dimensions as the space...for example, I have 3 vectors I am analyzing in a set in R3. Then I can simply take the determinant to figure out whether the set forms a basis for the vector space?

Answer 10

exactly!

Answer 11

That makes a lot of sense....thanks!

Answer 12

no problem!