Span and basis
The span of vectors is everything you can build from them by scaling and adding. A basis is an independent set that spans the whole space.
Last updated
Analogy
A basis is a minimal set of paint colours that can mix every shade you need.
Example
Any (2,3) = 2(1,0) + 3(0,1).Coordinates are 2 and 3.Another basis: (1,1) and (1,-1).Then (2,3) = 2.5(1,1) - 0.5(1,-1).
In practice
A basis gives every vector in the space a unique set of coordinates.
Practice
Flashcards
- What is a basis?An independent set of vectors that spans the whole space.
Interview questions
- Explain linear independence and test whether (1,2) and (2,4) are independent.
- Independent if the only combination giving zero is trivial
- (2,4) = 2(1,2)
- So 2(1,2) - (2,4) = 0 with non-zero coefficients
- Hence dependent
Questions people ask
Is a basis for a space unique?
No. Many different bases exist for the same space. In the plane, (1,0) and (0,1) form one, and (1,1) and (1,-1) form another. What stays fixed is the number of vectors in any basis, which is the dimension of the space.
How do I find coordinates in a new basis?
Write the vector as a combination of the basis vectors and solve for the coefficients. For (2,3) with basis (1,1) and (1,-1), solve a plus b equals 2 and a minus b equals 3. This gives a equals 2.5 and b equals -0.5.