# Span of R^3

• Mar 7th 2010, 09:27 AM
mulaosmanovicben
Span of R^3
I am having problems with this question could anyone help me?

Let u={u1,u2,u3}, v={v1,v2,v3} w={w1,w2,w3} be three vectors in R^3. Show that S={u,v,w} spans R^3 if an only if determinent of

u1 u2 u3
v1 v2 v3
w1 w2 w3

does not equal zero.

EDIT:

here is my attempt does it sound solid?

Since the determinent does not equal zero then the matrix is row equivalent to the Identity matrix. Then the augmented matrix

u1 u2 u3 a
v1 v2 v3 b
w1 w2 w3 c

can be written in reduced row echelon form and therefore S spans R^3
• May 22nd 2010, 08:37 PM
dwsmith
Quote:

Originally Posted by mulaosmanovicben
I am having problems with this question could anyone help me?

Let u={u1,u2,u3}, v={v1,v2,v3} w={w1,w2,w3} be three vectors in R^3. Show that S={u,v,w} spans R^3 if an only if determinent of

u1 u2 u3
v1 v2 v3
w1 w2 w3

does not equal zero.

EDIT:

here is my attempt does it sound solid?

Since the determinent does not equal zero then the matrix is row equivalent to the Identity matrix. Then the augmented matrix

u1 u2 u3 a
v1 v2 v3 b
w1 w2 w3 c

can be written in reduced row echelon form and therefore S spans R^3

If the det=0, then the vectors are lin. dep.

1 vector in \$\displaystyle \mathbb{R}^3\$ forms a line
2 vectors in \$\displaystyle \mathbb{R}^3\$ at most forms a plane

2 lin. ind. vectors can only span \$\displaystyle \mathbb{R}^2\$