# Thread: [SOLVED] is this the subspace?

1. ## [SOLVED] is this the subspace?

7.

v1 (1,0,1)
v2 (0,1,1)
v3 (-2,3,1)

a) show all 3 vectors are linearly independent. I calculated the det and it is 0 so they are all linearly independent.

I also found the zero vector, v3, which is v3 = 3*v2 - 2*v1

b) describe, geometrically, the subspace of r3 spanned by v1, v2 and v3.

c) find a vector w such that v1 and v2 and w are linearly independent.

this one i'm not sure about, my teacher said i could just calculate the cross product.

the cross product of v1 and v2 is (-1,-1,1). now what do i do with this? how do write the answer?

2. Originally Posted by thekrown
7.

v1 (1,0,1)
v2 (0,1,1)
v3 (-2,3,1)

a) show all 3 vectors are linearly independent. I calculated the det and it is 0 so they are all linearly independent.

It's exactly the other way around : iff the determinant of the matrix formed by the vectors is zero are these vectors lin. dependent ...and they

indeed are lin. dependent!

I also found the zero vector, v3, which is v3 = 3*v2 - 2*v1

b) describe, geometrically, the subspace of r3 spanned by v1, v2 and v3.

It's just the plane spanned by $\displaystyle v_1,v_2$ .

c) find a vector w such that v1 and v2 and w are linearly independent.

this one i'm not sure about, my teacher said i could just calculate the cross product.

Of course: the cross product of two vectors is a vector perpendicular to both vectors and, thus, lin. independent of them...

the cross product of v1 and v2 is (-1,-1,1). now what do i do with this? how do write the answer?

Just write that $\displaystyle v_1,v_2,(-1,-1,1)$ are lin. ind., and check this showing their determinant is not zero!

Tonio

.

3. Okay so to answer b I can simply write

the subspace of r3 is the planned by v1 and v2 and this is a sufficient answer?

4. Originally Posted by thekrown
Okay so to answer b I can simply write

the subspace of r3 is the planned by v1 and v2 and this is a sufficient answer?

How can anyone not taking that class know!? Perhaps your teacher wants to write explicitly the plane, or for him/her it is sufficient the above answer...

Tonio