Given the vectors (x1, y1) and (x2, y2), show that they are linearly independent if the qu...

Given the vectors (x1, y1) and (x2, y2), show that they are linearly independent if the quantity x1 y2−x2y1 is nonzero (see part (c) of Exercise). [Hint: Suppose x2 _= 0. If (x1, y1) and (x2, y2) are on the same line through (0, 0), then (x1, y1) = λ(x2, y2) for some λ. But then λ = x1/x2 and λ = y1/y2. What does this say about x1/x2 and y1/y2? What if x2 = 0?]

Exercise

Show that the vectors (x1, y1) and (x2, y2) are linearly dependent—that is, not linearly independent—if any of the following conditions are satisfied.

(a) If

(b) If  for some constant λ.

(c) If x1 y2 − x2 y1 = 0. Hint: Assume x1 is not zero; then  But , and we can use part b. The other cases are similar. Note that the quantity  is the determinant of the matrix