Thread: Vector Spaces

Let G be a vector space. Suppose that the span{x_1, ..., x_n} = G and suppose Y = {y_1, ..., y_m} is a set of lin. independent vectors in G. What are you able to say about the relationship between n and m? Why?

Originally Posted by Bloden

Let G be a vector space. Suppose that the span{x_1, ..., x_n} = G and suppose Y = {y_1, ..., y_m} is a set of lin. independent vectors in G. What are you able to say about the relationship between n and m? Why?

That,

Because any linear independant set can be enlarged into a basis.

That makes perfect sense, although seemed a little too simple for such a big homework problem. So, I asked my Prof. exactly what he was looking for, and he replied that he's looking for an explanation of why m > n, m < n, m = n and when any of them might happen, or when any of them might not happen. So I think he wants cases of why some of the others can not happen. You have that n >= m; is that always the case; and why will m > n never work?

Originally Posted by Bloden

Let G be a vector space. Suppose that the span{x_1, ..., x_n} = G and suppose Y = {y_1, ..., y_m} is a set of lin. independent vectors in G. What are you able to say about the relationship between n and m? Why?

The vector space has a dimension d, which is the number of vectors in a basis B.
For such a basis B, the d vectors span G and are also linearly independent, by definition.

Since X = {x_1...n} spans G, either n = d (then X is linearly independent) or n > d (then X is linearly depedent).
Since Y = {y_1...m} is linearly independent, m is at most d. If m = d, then Y is also a basis. If m < d, then Y doesn't span G.