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.

**i do not know how to do this. please help.** 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