Determine whether the set {[1 4], [-1 6], [3 8]} is a basis of R^2

To determine whether the set {[1 4], [-1 6], [3 8]} is a basis of R^2, we need to check two conditions:

1. Linear independence: If the vectors in the set are linearly independent, then the set could potentially be a basis of R^2.

2. Spanning: If the vectors in the set span R^2, then the set could potentially be a basis of R^2.

Let's go through these steps to check if the set is a basis of R^2:

1. Linear independence:
To check for linear independence, we need to solve the equation c1 * [1 4] + c2 * [-1 6] + c3 * [3 8] = [0 0].

This equation can be written as the matrix equation:

[[1 -1 3] [4 6 8]] * [c1 c2 c3] = [0 0].

We can solve this system of linear equations by row reducing the augmented matrix:

[1 -1 3 | 0]
[4 6 8 | 0].

After performing Gaussian elimination, the row-reduced echelon form of the augmented matrix is:

[1 0 0 | 0]
[0 1 2 | 0].

Since the system has a non-trivial solution (c1 = 0, c2 = -2, c3 = 1), we can conclude that the vectors in the set are linearly dependent, and thus, the set {[1 4], [-1 6], [3 8]} is not linearly independent.

Therefore, the set {[1 4], [-1 6], [3 8]} cannot be a basis of R^2, as it fails the condition of linear independence.