Let u 2 2 2 v 2 5 3 and w 0 7 h a List all the values of

Let u = [2 2 -2], v = [-2 5 -3] and w = [0 7 h]. (a) List all the values of h for which {u, v, w} spans R^3. Justify your answer. (b) List the values of h so that {u, v, w} forms a linearly dependent set. Justify your answer. (c) For the values of h (b), write w as a linear combination of u and v.

Solution

4. Let A be the matrix with the given vectors as columns i.e. let A =

2

-2

0

2

5

7

-2

-3

h

To answer the given questions, we will reduce A to its RREF as under:

1.Multiply the 1st row by ½

2.Add -2 times the 1st row to the 2nd row

3.Add 2 times the 1st row to the 3rd row

4.Multiply the 2nd row by 1/7

5.Add 5 times the 2nd row to the 3rd row

6.Multiply the 3rd row by 1/(h+5)

7.Add -1/(h+5) times the 3rd row to the 2nd row

8.Add 1 times the 2nd row to the 1st row

Then the RREF of A is I₃ i.e.

1

0

0

0

1

0

0

0

1

(a). The given set of vectors i.e. {u,v,w} will span R³ if the RREF of A is I₃, i.e. h -5. ( in the 6^th row-operation, we have multiplied the 3rd row by 1/(h+5). Since division by 0 is not defined, this is possible only if h -5).

(b).If h +5 = 0, i.e. if h = -5, then we have (0,7,-5)^T = (2,2,-2)^T+(-2,5,-3)^T. Then the given set {u,v,w} becomes linearly dependent.

(c ). If h = -5, then we have (0,7,-5)^T = (2,2,-2)^T+(-2,5,-3)^T i.e. w = u+v.

2	-2	0
2	5	7
-2	-3	h