What is the value of t that would result in span((0, 1, −1),(1, 2, 1),(t, −1, 4))

Question

Determine which values of t ∈ R result in span((0, 1, −1),(1, 2, 1),(t, −1, 4)) being

(i) a line

(ii) a plane

or (iii) all of R³

Answer 1

{(0, 1, -1), (1, 2, 1)} is a linearly independent set. (When there are only two vectors, it's simple: linearly dependent if one is a scalar multiple of the other; linearly independent otherwise. One warning: for more than two vectors, it's not quite that simple.)

(i) There's no value of t for which the span is a line- the argument above shows the span is at least 2-dimensional.

(ii) For t=1, span is a plane. (1, -1, 4) = (1, 2, 1) - 3 * (0, 1, -1)

To figure out that the three vectors are linearly dependent when t=1, I computed the determinant of the matrix with rows (t, -1, 4), (1, 2, 1) and (0, 1, -1). The determinant is -3 + 3t, which is 0 when t = 1

(iii) For t≠1, the determinant of the matrix is not zero. That means the three vectors are independent and their span is therefore R³.