Let A 3 1 5 B 3 4 4 and C 9 6 13 1 Determine whether or n

Let A= -3 -1 -5 , B= 3 4 4 , and C= 9 6 13 . 1. Determine whether or not the three vectors listed above are linearly independent or linearly dependent. If they are linearly dependent, determine a non-trivial linear relation. Otherwise, if the vectors are linearly independent, enter 0\'s for the coefficients, since that relationship

Solution

To determine, whether the vectors are linear independent or not we will use the definition.

Three vectors V1, V2 and V3 are linearly independent if the only solution to

A * V1 + B * V2 + C * V3 = 0 has Zero solution

That is, . A = 0, B = 0 and C = 0.

A = [-3 -1 -5]

B = [3 4 4]

C = [9 6 13]

Suppose,

C = kA + lB

We get 3 equations from here...

-3k + 3l = 9 ...1
-k + 4l = 6 ...2
-5k + 4l = 3 ...3

Subtract equation 3 from equation 2, we get..

-k + 4l = 6   
-(-5k + 4l = 3)

=> -k + 5k + 4l- 4l = 6 - 3

=> 4k= 3

=> k= 3/4

Plug this in equation 2

-3/4 + 4l = 6

=> 4l = 6 + 3/4 = 24/4 + 3/4 = 27/4

=> l = 27/16

Now we need to plug the values of k and l in equation 1 to see whether the vectors are dependent or independent.

-3k + 3l = -3(3/4) + 3(27/16) = -9/4 + 81/16 = -36/16 + 81/16 = 45/16

as 45/16 is not equal to 9, so the vectors are linearly independent.

Therefore, 0*A+0*B+0*C=0