Suppose v1 minus1 minus3 4 v2 2 7 minus7 v3 minus5 0 h Fo

Suppose v_1 = [minus1 minus3 4], v_2 = [2 7 minus7], v_3 = [minus5 0 h] For what values of h is v_3 in span {v_1, v_2}? For what values of h is in span {v_1, v_2, v_3} linearly independent? Determine the values of k such that the matrix is the augumented matrix of a consistent system. [1 minus2 k 4 5 minus3]

Solution

Post one more question to get the answer to problem number 5

Q4)

For the vector v3 to be in span of v1 and v2

a) v3 = av1 + bv2

(-5,0,h) = a(-1,-3,4) + b(2,7,-7)

-5 = -a + 2b

0 = -3a + 7b

b = 3a/7

-5 = -a + 6a/7

a = 35

b = 15

h = 35(4) + 15(-7)

=> 140 - 105 = 35

Hence the value of h must be 35 in order to be in the span of v1 and v2

b)

The matrix formed by v1,v2 and v3 must be non-zero

Determinant = -1(7h) + 2(3h) -5(21+28)

Determinant = -h + 35

Hence the value of h must not be equal to 35

-1	2	-5
-3	7	0
4	-7	h