Step by step
The question
Question: are (1, 2, 3) and (0, 1, 1) a basis for R³?
v1 = [1; 2; 3], v2 = [0; 1; 1]
A basis for R³ is a set of vectors that are independent, and that make every vector of R³ as a linear combination.
basis for R³: independent, and every vector
is a linear combination of them
1 Count the vectors
There are 2 vectors, but every basis for R³ has exactly 3.
2 vectors cannot reach every vector of R³.
2 vectors, but R³ needs 3
So they are not a basis.
= not a basis
2 Check
Let's check with one vector: (1, 0, 0).
We try to make it from them.
The vectors as columns, then this one after the bar.
= [1, 0, 1; 2, 1, 0; 3, 1, 0]
Row reduced, as before.
Row 3 says 0 = −1: false.
→ [1, 0, 1; 0, 1, −2; 0, 0, −1]
So (1, 0, 0) cannot be made from them.
They do not reach every vector: the answer is right!
Let's look back.
First: count the vectors.
Last: check.
Answer
= not a basis