S span u v S Let S span u v For each of the following sets

S = span {u, v} S Let S = span {u, v}. For each of the following sets of vectors determine whether S is a line or a plane. u = [-1 -2 -5], v = [-4 -11 -21], u = [2 3 2], v = [-4 -6 -4], u = [3 -10 -5], v = [0 0 0], u = [-3 -5 -1], v = [-8 -15 -5]

Solution

The set would be a plane if the vectors are linearly independent and a line if they are linearly dependent

For non zero two vectors to be linearly dependent one must be a multiple of the other

1.

Vectors are not multiples of each other hence linearly independent

Hence span S is a plane.

2.

-2u=v

So vectors are linearly dependent. HEnce span S is a line

3.

Second vector is a 0 vector

Hence span S is a line

4.

Vectors are not multiples of each other. Hence Span S is a plane