Textbook Guide Mathematics: Euclidean Vector, Row And Column Spaces, Symmetric Matrix
Document Summary
A vector space is the set (cid:1848) composed of vectors. Sum of two vectors (cid:2203) and (cid:2204) is shown as (cid:2203)+(cid:2204). The multiplication of (cid:2203) by the scalar (cid:1855) is shown as (cid:1855)(cid:2203) that is also in the set (cid:1848). If is a subset of (cid:1848), then it should satisfy the properties: The zero vector of (cid:1848) is in subset. Is closed under vector addition: for all (cid:2203) and (cid:2204) vectors in , (cid:2203)+(cid:2204) is also in . Is closed under vector multiplication by scalars: for all (cid:2203) vectors in and scalars (cid:1855), (cid:1855)(cid:2203) is also in . Linear combination: sum of any scalar multiples of vectors. Span{(cid:2204)(cid:2778), ,(cid:2204)(cid:2198)}: set of all vectors that can be written as linear combinations of (cid:2204)(cid:2778), ,(cid:2204)(cid:2198). Span{(cid:2204)(cid:2778), ,(cid:2204)(cid:2198)} is a subspace of (cid:1848) provided that (cid:2204)(cid:2778), ,(cid:2204)(cid:2198) are in the vector space (cid:1848). Span{(cid:2204)(cid:2778), ,(cid:2204)(cid:2198)} is called the subspace spanned/generated by (cid:2204)(cid:2778), ,(cid:2204)(cid:2198).
