Example · High school
Terms
| Term | This example | Why |
|---|
| basis |
✓ |
A basis need not be perpendicular: $(x, y) = (x - y)(1, 0) + y (1, 1)$.
|
| linearly independent |
✓ |
|
| spanning set |
✓ |
|
| pairwise linearly independent |
✓ |
The vectors $(1, 0)$ and $(1, 1)$ are pairwise linearly independent, because neither vector is a scalar multiple of the other.
|
| orthogonal |
✗ |
The vectors $(1, 0)$ and $(1, 1)$ are not orthogonal, because their dot product is $1 \cdot 1 + 0 \cdot 1 = 1$.
|
| orthonormal |
✗ |
The vectors $(1, 0)$ and $(1, 1)$ are not orthonormal, because their dot product is $1$ and $(1, 1)$ has length $\sqrt{2}$.
|
Distinctions using this example
- Linearly independent vs. orthogonal: The vectors $(1, 0)$ and $(1, 1)$ show that linearly independent vectors need not be orthogonal: the vectors point in different directions, but not at a right angle.
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