Determine if the following vectors are linearly independent

Determine if the following vectors are linearly independent. v_l = [4, minus5, 2, 6], v_2 = [2, minus2, 1, 3], v_3 = [6, minus3, 3, 9], v_4 = [4, minus1, 5, 6] Let M =[4    2    6    4 minus5 minus2 minus3 minus1 2 1 3 5 6 3 9 6] Using octave, find the inverse of M. (Verify your answer by multiplying M and inv(M) together.) Record your results here. Find the determinant of M. The rank of a matrix is the number of leading variables in the general solution of Ax = 0. Octave includes a rank function which returns the rank of a matrix. The rank must be 4 if these are linearly independent. What is rank(M)? What can you conclude?

Solution

they are not linearly independent

v_3 = 9 v_2 - 3 v_1

Octave

X =

4 2 6 4
-5 -2 -3 -1
2 1 3 5
6 3 9 6

Y = inv(X)
Warning: Matrix is singular to working precision.

Y =

Inf Inf Inf Inf
Inf Inf Inf Inf
Inf Inf Inf Inf
Inf Inf Inf Inf

Y = pinv(X)

Y =

-0.0659 -0.3077 0.1099 -0.0989
-0.0073 -0.0769 0.0037 -0.0110
0.1319 0.2308 -0.2967 0.1978
-0.0513 0.0000 0.3333 -0.0769

>> Y*X

ans =

0.9011 0.2967 -0.0330 0
0.2967 0.1099 0.0989 0.0000
-0.0330 0.0989 0.9890 0.0000
-0.0000 -0.0000 -0.0000 1.0000

>> X*Y

ans =

0.3077 0.0000 0.0000 0.4615
-0.0000 1.0000 0.0000 -0.0000
-0.0000 0.0000 1.0000 0.0000
0.4615 0.0000 0 0.6923

determinant of M = 0

rank(X) = 3

matrix is not invertible