Are these vectors linearly independent or dependent ?

Question

( 1, -1, 2 ) ( 2,0,1) ( -1, 2, -1  ) 

 

Is there a way to figure it out by looking at them instead of the matrix way ? 

Answer 1

If there were only 2 vectors, they would be linearly independent if one is not a multiple of the other.  But since you have 3 vectors, it is not so obvious.  Evaluate the determinant of the matrix.  If it gives zero, the vectors are linearly dependent.  If it does not give zero, they are linearly independent.

Answer 2

Not really. You have to prove that none of them are a linear combination of the others

by trial and error.

 

1    -1    2

2    0      1

-1   2     -1

 

STEP 1:  row2 = -2*row 1 + row2;  row3 = row 1 + row3

1    -1    2

0     2    -3

0     1     1

 

 

STEP 2: row2 = -2*row3 + row2

1   -1    2

0   0     -5 

0   1      1

 

 

STEP 3:  row2 = row2/-5

1   -1    2

0    0    1

0    1    1

 

 

 

STEP 4:   row1 = -row3 + row1

1   0   3

0   0   1

0   1   1

 

STEP 5:  row1  = -3*row2 + row1

1  0   0

0  0   1

0  1   1

 

 

STEP 6:  row3 = -row2 + row3

1   0   0

0  0    1

0  1    0

 

 

Swapping rows 2 and 3 produce the identity matrix.