Lecture 14 VECTOR SPACES Informal definition A vector

Lecture 14 VECTOR SPACES

Informal definition A vector space is a collection of objects that one can add and scale (multiply by numbers), together with requirements: The zero, as usual, does nothing: Very informally: a vector space is just a batch of objects that behave like vectors.

Examples of vector spaces Example 1. Vectors themselves form a vector space. One can indeed add and scale vectors. The zero vector is the zero of the vector space. Example 2. Functions.

Examples of vector spaces (continued) The sum of two polynomials of degree 2 may not be a degree 2 polynomial (or any other matrices of fixed size) Example 6. All integers do not form a vector space. The sum of integers is obviously an integer; The zero would be the zero.

Subspaces The same abstract definition as we saw a week ago. Example 1.

Subspaces (continued) General fact: Any subspace is a vector space on its own. • The sum of two symmetric matrices is symmetric; • A scaling of any symmetric matrix is a symmetric matrix.

Linear combinations Since in a vector space we can add and scale objects, linear combinations make perfect sense there.

Spans Since we have linear combinations, we also have spans.

Spans: examples Let’s unravel and combine the like terms: !!! Thus,

Spans: examples Using algebra of matrices, !!! Two matrices are equal if they are equal entrywise.