Edwards-Penney Chapter 4, sections 4.1, 4.2, 4.3, 4.4 Topics, Definitions, Theorems 4.1 ==== DEF. Vector DEF. Vector add, scalar multiply DEF. Vector Spaces R^2 and R^3 AXIOMS for a vector space DEF. Linear independence and dependence Two vectors in R^2 Three vectors in R^3 Determinant test DEF. Span DEF. Basis: span plus independent DEF. Standard basis, the columns of the identity. THEOREM. Three independent vectors in R^3 are a basis for R^3. SUBSPACE: A nonvoid subset of a vector space V that under the same add and scalar multiply operations is itself a vector space. SUBSPACE CRITERION: A nonvoid set S in a vector space V closed under the add and scalar multiply of V is itself a vector space, hence a subspace of V. 4.2 ==== DEF. Vector space R^n DEF. Add and scalar multiply of n-vectors AXIOMS for an abstract vector space 4 rules for add 4 rules for scalar multiply DEF. Vector space of functions DEF. A subspace W is a nonvoid subset of a vector space V which is itself a vector space using the same add and scalar multiply. THEOREM 1. [Subspace criterion] A set W is a subspace of vector space V provided (a) the zero vector is in W (W is nonvoid), (b) u,v in W ==> u+v is in W, (c) c a scalar and u in W ==> cu is in W. EXAMPLE. W is the subset of V=R^n satisfying a linear homogeneous algebraic equation. Then W is a subspace of R^n. EXAMPLE. W is the subset of R^4 with all coordinates nonnegative. Then W is not a subapce of R^4. EXAMPLE. W is a subset of R^4 defined by x1*x4=0. then W is not a subspace of R^4. THEOREM 2. Let W be the subset of R^n consisting of all solutions x to a homogeneous matrix equation Ax=0. Then W is a subspace of R^n. DEF. Trivial subspaces 0 and R^n. Proper subspace. EXAMPLE. Given a linear homogeneous 3x4 system, briefy Ax=0. For the example, the vector form of the general solution has a basis of two vectors u,v. The solution space W of the equation Ax=0 (the KERNEL of matrix A) is then 2-dimensional and W = set of all x = t1 u + t2 v where t1, t2 are arbitrary constants. 4.3 ==== DEF. Linear combination DEF. Span, spanning set THEOREM 1. The span(a set of vectors) is a subspace of V for any anstract vector space V. DEF. Linear independence in an abstract vector space DEF. Standard unit vectors in R^N are the columns of the identity matrix. EXAMPLE. Apply the definition of independence to 3 vectors in R^4. Independence test reduces to Ax=0 has unique solution x=0. DEF. Linear dependence in an abstract vector space. THEOREM. A set of more than n vectors in R^n is linearly dependent. THEOREM. A set of vectors is dependent if and only if one of them is a linear combination of the others. THEOREM 2. [Determinant Test] A set of n vectors in R^n is INDEPENDENT if and only if the determinant of their augmented matrix is nonzero. A set of n vectors in R^n is DEPENDENT if and only if the determinant of their augmented matrix is zero. THEOREM 3. [Rank Test] A set of m vectors in R^n with m=n is independent if and only if the rank of the augmented matrix of the vectors is equal to its row dimension (which is m). 4.4 ==== DEF. Basis. A finite set S of vectors in an abstract vector space V is called a basis for V provided (a) S is a set of independent vectors, (b) V=span(S). EXAMPLE. The standard basis, the columns of the identity, is a basis for R^n. THEOREM. A set of n independent vectors in R^n is a basis for R^n. A list of vectors containing the zero vector is dependent. THEOREM 1. If n vectors form a basis for vector space V, then any n+1 vectors in V are dependent. THEOREM 2. Any two bases for an abstract vector space V must have the same number of elements. DEF. The number of elements in a basis for V is called the dimension of V, written dim(V). EXAMPLE. Polynomials P_n of degree n or less is a vector space of dimension n+1. EXAMPLE. All polynomials P is a vector space of infinite dimension. THEOREM 4. The following results hold. THEOREM. If dim(V)=n and S is a set of n vectors in V that spans V, then S is independent and S is a basis for V. THEOREM. If dim(V)=n and S is a set of n independent vectors in V, then S already spans V and S is a basis for V. THEOREM. If dim(V)=n and S is any set of independent vectors in V, then V has a basis which contains the vectors in S. THEOREM. If dim(V)=n and S is a set of vectors which spans V, then S contains a basis for V. ALGORITHM. How to construct a basis for the solution space W of a matrix equation Ax=0. 1. Find rref(A), which is the Last combo-swap-mult Frame. 2. Apply the Last Frame Algorithm to find the scalar general solution. Construct from it the vector general solution with invented symbols t1, t2, t3, etc. 3. A basis for W is obtained by computing the vector partial derivatives on the symbols. These same vectors can be found by setting one symbol equal to 1 and the rest equal to 0, in all possible ways. EXAMPLE. Find a basis for the solution space of a 3x5 linear homogeneous system Ax=0. There are 3 invented symbols. A basis of 3 vectors is obtained by taking partial derivatives on the 3 symbols.