Linear algebra review
نویسنده
چکیده
For each natural number n, the n-dimensional Euclidean space is denoted by R, and it is a vector space over the real field R (i.e., R is a real vector space). Vectors v1, . . . , vk ∈ R are linearly dependent if there exist c1, . . . , ck ∈ R, not all zero, such that c1v1 + · · ·+ ckvk = 0. If v1, . . . , vk ∈ R are not linearly dependent, then we say they are linearly independent. The span of v1, . . . , vk, denoted by span{v1, . . . , vk}, is the space of all linear combinations of v1, . . . , vk, i.e., span{v1, . . . , vk} = {c1v1 + · · ·+ ckvk : c1, . . . , ck ∈ R}. The span of a collection of vectors from R is a subspace of R, which is itself a real vector space in its own right. If v1, . . . , vk ∈ R are linearly independent, then they form an (ordered) basis for span{v1, . . . , vk}. In this case, for every vector u ∈ span{v1, . . . , vk}, there is a unique choice of c1, . . . , ck ∈ R such that u = c1v1 + · · ·+ckvk. We agree on a special ordered basis e1, . . . , en for R, which we call the standard coordinate basis for R. This ordered basis defines a coordinate system, and we write vectors v ∈ R in terms of this coordinate system, as v = (v1, . . . , vn) = ∑n i=1 viei. The (Euclidean) inner product (or dot product) on R will be written either using the transpose notation, uv, or the angle bracket notation, 〈u, v〉. In terms of their coordinates u = (u1, . . . , un) and v = (v1, . . . , vn), we have
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