Question

How do you show a set of vectors span? What do you have to prove?

Answer 1

Theorem: A set of vectors B={w1, w2, ... w(k)} in a vector space W is a basis for W if and only if the set B is linearly independent AND spans W.

Answer 2

Showing linearly independent I got... but how do I show it spans??

the vectors are u1=(1,1,0), u2=(0,1,0), and u3=(-1,0,1)

Answer 3

The question you're asking isn't complete.

"how do you show a set of vectors spans.. (a vector space -- like \(\mathbb{R}^3\) or \(\mathbb{R}^4\), etc.)"

The answer to that is to count pivots. The vectors you'e given can be viewed as columns of a matrix. Row reduce that matrix, and count the pivots. :)

Answer 4

Well the problem in the book states this.
Show that U={u1, u2, u3} where u1=(1,1,0), u2=(0,1,0), u3=(-1,0,1) is a basis for R3.

In order to show that a set of vectors is a basis, the theorem states that theorem that I listed above. Now I showed they are linearly independent, it the theorem also states you must show that it spans U as well.

How would counting pivots verify spanning?

Answer 5

If you have 3 linearly independent vectors, you have 3 pivots. 3 linearly independent vecotrs in R3 form a basis for R3.