Step by step
The question
Question: are (1, 2), (3, 4) and (5, 6) a basis for R²?
v1 = [1; 2], v2 = [3; 4], v3 = [5; 6]
A basis for R² is a set of vectors that are independent, and that make every vector of R² as a linear combination.
basis for R²: independent, and every vector
is a linear combination of them
1 Count the vectors
There are 3 vectors, but every basis for R² has exactly 2.
3 vectors in R² cannot be independent.
3 vectors, but R² needs 2
So they are not a basis.
= not a basis
2 Check
Let's check: they must be dependent.
Row reducing, as in the independence lesson, gives v₁ − 2v₂ + v₃ = 0.
= [1; 2] − 2 × [3; 4] + [5; 6]
= [1; 2] − [2 × 3; 2 × 4] + [5; 6]
= [1; 2] − [6; 8] + [5; 6]
= [1 − 6 + 5; 2 − 8 + 6]
= [0; 0]
We get the zero vector, and not all the numbers are 0.
So they are dependent: the answer is right!
Let's look back.
First: count the vectors.
Last: check.
Answer
= not a basis