Vector space
A set of vectors you can add together and scale by numbers, while always staying inside the set. A subspace is a vector space sitting inside a bigger one.
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Analogy
A flat sheet of paper extending forever: any two arrows on it add to another arrow on it.
Example
The line y = 2x is a subspace.(1,2) + (2,4) = (3,6), still on it.The line y = 2x + 1 is not.It misses the origin (0,0).
In practice
Gives one framework covering arrows, polynomials, functions and data.
Practice
Flashcards
- Why must a subspace contain the zero vector?Scaling any member by 0 gives the zero vector.
Questions people ask
Why must a subspace contain the zero vector?
Scaling any vector in the subspace by zero must stay inside it, and that gives the zero vector. So a line that misses the origin, like y equals 2x plus 1, cannot be a subspace. This is a quick first test. Always check the zero vector before testing anything else.
Are polynomials a vector space?
Yes. Adding two polynomials gives a polynomial, and multiplying by a number gives one too. So polynomials of degree at most 2 form a vector space with 3 dimensions, built from 1, x and x squared. Functions and sequences can also form vector spaces.