Find the values of h for which the vectors 2 4 1 6 7 3 and 8

Find the value(s) of h for which the vectors (2 -4 1), (-6 7 -3), and (8 h 4) are R-linearly independent. 2. Find the value(s) of k for which the vectors (1 -1 3), (-5 7 8), and (1 1 k) are R-linearly dependent.

Solution

1.

Let

a(2,-4,1)\'+b(-6,7,-3)\'+c(8,h,4)\'=0

2a-6b=8

a-3b=4 ,a=3b+4

-4a+7b=h

-4(3b+4)+7b=h

-15b-16=h

-15b=-h+16

For linear indepdendence we need b=0 hence, h=16

So h must be 16

2.

Let,

a(1,-1,3)+b(-5,7,8)+c(1,1,k)=0

a-5b+c=0

a+7b+c=0

3a+8b+kc=0

First two equation give b=0

So,a=-c

3a+kc=0

-3c+kc=0

(k-3)c=0

So for k=3 , c can take any values

Hence, k=3 for the vectors to be linearly independent.