What are the values for a and b?

Question

If k� is a real number, then the vectors (1,k),(k,6k+7)(1,�),(�,6�+7) are linearly independent precisely when k≠a,b�≠�,�,

where a=�= , b=�=  , and a<b�<�.

Answer 1

The vectors are linearly independent when their determinant is not zero: =6k+7−k^2

Set it equal to zero to find when they are not independent:

6k+7−k^2=0

k^2−6k−7=0

(k−7)(k+1)=0

So:

k=−1ork=7

Therefore,

a=−1,b=7

​and the vectors are linearly independent precisely when:

k≠−1