Describe geometrically line plane or all of R3 all linear co

Describe geometrically (line, plane, or all of R^3) all linear combinations of (a) [1 2 3] and [3 6 9] (b) [1 0 0] and [0 2 3] (c) [2 0 0] and [0 2 2] and [2 2 3]

Solution

All linear combinations of the vectors is the span of the vectors .

for 1) we see that (3,6,9) is a multiple (namely- 3) of the vector (1,2,3)- hence the two vectors are linearly

dependent, and the span of these 2 vectors are a line in R3

2) As neither vector is a scalar multiple of the other, we see they are linearly independent and hence the span of

these 2 vectors will be a 2-dimensional plane in R3

3) We must do a little bit more work than the previous 2 as we have 3 vectors. Hence just form a matrix with each

column a vector from the set you were provided, do row reduction and see if you get a pivot in all 3 columns of the

row reduced form of the matrix- if we do the 3 vectors are linearly independent- which in this case they are- so

they span all of R3 as we have 3 linearly independent vectors form R3 which means they must necessarily form a

basis for R3.