To determine the possible echelon forms of the matrix A, we need to consider the conditions given in the problem statement.
The matrix A is a 4x3 matrix, so it has 4 rows and 3 columns. Let's denote the elements of A as follows:
A = [a1 a2 a3]
According to the problem statement, the set {a1, a2} is linearly independent, which means that neither a1 nor a2 can be expressed as a linear combination of the other. Additionally, a3 is not in the span of {a1, a2}, which means that a3 cannot be written as a linear combination of a1 and a2.
Now, let's consider the possible echelon forms of A.
Possible Echelon Forms of A:
1. [a1 a2 a3]
[0 0 *]
[0 0 0]
[0 0 0]
In this echelon form, the first row contains the linearly independent vectors a1 and a2. The second row is all zeros because we cannot express a3 as a linear combination of a1 and a2. The third and fourth rows are also all zeros.
2. [a1 a2 0]
[0 0 *]
[0 0 0]
[0 0 0]
In this echelon form, the first row contains the linearly independent vectors a1 and a2. The third row is all zeros because a3 is not in the span of a1 and a2. The second and fourth rows are also all zeros.
Note: The * in the echelon forms represents any value, as it is not constrained.
Now, let's move on to part (a) of the problem.
(a) To determine if the set {v1, v2, v3} is linearly independent, we need to check if there exists a non-trivial linear combination of these vectors that equals the zero vector.
Let's denote the vectors as follows:
v1 = a1
v2 = a2
v3 = a3
To check for linear dependence, we set up the following equation:
c1*v1 + c2*v2 + c3*v3 = 0
where c1, c2, and c3 are constants.
Since v1 and v2 are linearly independent, we know that c1 = c2 = 0. Thus, the equation becomes:
c3*v3 = 0
Since a3 is not in the span of a1 and a2, we can conclude that c3 must be equal to zero. Therefore, the set {v1, v2, v3} is linearly independent.
Now, let's move on to part (b) of the problem.
(b) Since the set {v1, v2, v3} is linearly independent, there is no linear dependence relation among these vectors.