Decide whether the following vectors are linearly independent or not (on ()
a) ((, 0) and (0, 1)
b) (-1, 1, 0) and (0, 1, 2)
c) (0, 1, 1), (0, 2, 1), and (1, 5, 3)
a) The vectors (1, 0) and (0, 1) are linearly independent because you cannot multiply one vector by a scalar to obtain the other vector.
b) The vectors (-1, 1, 0) and (0, 1, 2) are linearly independent because you cannot multiply one vector by a scalar to obtain the other vector.
c) The vectors (0, 1, 1), (0, 2, 1), and (1, 5, 3) are linearly independent. One way to prove this is to form a matrix with the vectors as columns and row reduce it. If the row-reduced matrix has a row of zeros, then the vectors are linearly dependent. However, in this case, the row-reduced matrix does not have a row of zeros. Therefore, the vectors are linearly independent.
To decide whether the given vectors are linearly independent or not, we can use the following steps:
a) ((0, 0) and (0, 1):
Step 1: Form the linear combination of the vectors:
c1(0, 0) + c2(0, 1) = (0, 0)
Step 2: Set up a system of equations and solve for the coefficients (c1, c2):
0*c1 + 0*c2 = 0
0*c1 + 1*c2 = 0
The system of equations simplifies to 0 = 0 and 0 = 0. Since the solution is c1 = c2 = 0, the vectors are linearly independent.
b) (-1, 1, 0) and (0, 1, 2):
Step 1: Form the linear combination of the vectors:
c1(-1, 1, 0) + c2(0, 1, 2) = (0, 0, 0)
Step 2: Set up a system of equations and solve for the coefficients (c1, c2):
-1*c1 + 0*c2 = 0
1*c1 + 1*c2 = 0
0*c1 + 2*c2 = 0
The system of equations simplifies to -c1 = 0, c1 + c2 = 0, and 2c2 = 0. Since the solution is c1 = c2 = 0, the vectors are linearly independent.
c) (0, 1, 1), (0, 2, 1), and (1, 5, 3):
Step 1: Form the linear combination of the vectors:
c1(0, 1, 1) + c2(0, 2, 1) + c3(1, 5, 3) = (0, 0, 0)
Step 2: Set up a system of equations and solve for the coefficients (c1, c2, c3):
0*c1 + 0*c2 + 1*c3 = 0
1*c1 + 2*c2 + 5*c3 = 0
1*c1 + 1*c2 + 3*c3 = 0
The system of equations simplifies to c3 = 0, c1 + 2c2 = 0, and c1 + c2 + 3c3 = 0. Since the solution is c1 = c2 = c3 = 0, the vectors are linearly independent.
To summarize:
a) ((0, 0) and (0, 1) are linearly independent.
b) (-1, 1, 0) and (0, 1, 2) are linearly independent.
c) (0, 1, 1), (0, 2, 1), and (1, 5, 3) are linearly independent.