Show that the set {[2,3],[4,1]} is linearly independent
To show that the set {[2,3],[4,1]} is linearly independent, we need to prove that the only solution to the equation x[2,3] + y[4,1] = [0,0] is the trivial solution x = 0 and y = 0.
Let's write out the equation using the components of the vectors:
x[2,3] + y[4,1] = [0,0]
(2x, 3x) + (4y, y) = (0,0)
Now we can split the equation into two separate equations for each component:
2x + 4y = 0
3x + y = 0
To solve this system of equations, we can use any method we prefer, such as substitution or elimination. Let's use elimination:
Multiply the second equation by 2:
6x + 2y = 0
Now we can subtract this new equation from the first equation:
(2x + 4y) - (6x + 2y) = 0 - 0
-4x + 2y = 0
Divide this equation by -2:
2x - y = 0
Now we have a new system of equations:
-4x + 2y = 0
2x - y = 0
Let's solve this system using elimination again:
Multiply the second equation by 2:
4x - 2y = 0
Add this new equation to the first equation:
(-4x + 2y) + (4x - 2y) = 0 + 0
0 = 0
Since the result is always true, this means that the system of equations has infinitely many solutions. Therefore, there are infinitely many solutions to the equation x[2,3] + y[4,1] = [0,0], and the vectors [2,3] and [4,1] are linearly dependent.
Hence, the set {[2,3],[4,1]} is not linearly independent.