Is this set a subspace of R3 or not?
Explain why or why not.
Span{[1 2 1],[-1 1 3]}
I've tried to do a three variable three unknown equation to solve for the scalars for each of the vectors but when doing it got very wrong numbers.
To determine whether the set is a subspace of ℝ³ or not, we have to check two conditions:
1. Closure under addition: For any two vectors in the set, their sum must also be in the set.
2. Closure under scalar multiplication: For any vector in the set, the product of the vector and any scalar must also be in the set.
Let's examine each condition:
1. Closure under addition: Consider two vectors in the set: [1 2 1] and [-1 1 3]. To check closure under addition, we need to find the sum of these vectors:
[1 2 1] + [-1 1 3] = [0 3 4]
Is [0 3 4] in the set? Since [0 3 4] cannot be formed as a linear combination of [1 2 1] and [-1 1 3] (you mentioned that you obtained incorrect numbers when solving the system of equations), we can conclude that the set does not satisfy closure under addition.
Since the set fails the closure under addition condition, it cannot be a subspace of ℝ³.
It's worth noting that to ensure closure under addition, the sum of any two vectors in the set needs to be a linear combination of those vectors.
Therefore, the set Span{[1 2 1],[-1 1 3]} is not a subspace of ℝ³.