MATH 110 Module 7 Exam | MATH110 Module 7 Exam Introduction to Statistics Latest 2024 | Portage Learning

MATH 110 Module 7 Exam | MATH110 Module 7 Exam Introduction to Statistics Latest 2024 | Portage Learning linear combination (2.3) - answersA linear combination of a list v1,....,vm of vectors in V is a vector of the form: a1v1+....+amvm where a1,...,am are in F. Span - answersThe set of all linear combinations of a list of vectors v1,....,vm in V is called the span of v1,....,vm, denoted span(v1,.....,vm). In other words, span(v1,....,vm) = {a1v1+...+amvm : a1,....,am} The span of a list of vectors in V is the smallest.... - answersSpan is the smallest containing subspace. The span of a list of vectors in V is the smallest subspace of V containing all the vectors in the list. spans - answersIf span(v1,....,vm) equals V, we say that v1,...,vm spans V. finite-dimensional vector space - answersA vector space is called finite-dimensional if some list of vectors in it spans the space. Polynomial P(F) - answersA function p:F-F is called a polynomial with coefficients in F if there exist a0,...,am in F such that p(z)= a0+a1z+a2z^2+.....+ amz^m for all z in F. P(F) is the set of all polynomials with coefficients in F. degree of a polynomial, deg p - answersA polynomial p in P(F) is said to have degree m if there exist scalars a0,a1,...,am in F with am not equal 0 such that p(z)= a0+a1z+...+amz^m for all z in F. -If p has degree m, we write degp =m. Pm(F) - answersFor m a nonnegative integer, Pm(F) denotes the set of all polynomials with coefficients in F and degree at most m. infinite-dimensional vector space - answersA vector space is called infinite-dimensional if it is not finite-dimensional linearly independent - answersA list v1,....,vm of vectors in V is called linearly independent if the only choice of a1,...,am in F that makes a1v1+...+amvm= 0 is a1=....=am=0 The empty list {0} is also declared to be linearly independent. linearly dependent - answersA list of vectors in V is called linearly dependent if it is not linearly independent. In other words, a list v1,...,vm of vectors in V is linearly dependent if there exist a1,...,am in F, not all 0, such that a1v1+...+amvm=0. Suppose v1,...,vm is a linearly dependent list in V. Then there exists j in {1,2,...m} such that the following hold: - answers(a) vj is in span(v1,...,vj-1) (b) if the jth term is removed from v1,...,vm, the span of the remaining list equals span(v1,...,vm) Length of linearly independent list is______________ the length of spanning list less than greater than equal too less than and equal too - answersLength of linearly independent list less than and equal toothe length of spanning list In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors. Every subspace of a finite-dimensional vector space is - answersEvery subspace of a finite-dimensional vector space is finite-dimensional basis - answersA basis of V is a list of vectors in V that is linearly independent and spans V. Criterion for basis - answersA list v1,...,vn of vectors in V is a basis of V if and only if every v in V can be written uniquely in the form: v = a1v1+...+anvn where a1,...,an is in F. Every spanning list in a vector space can be reduced... - answersSpanning list contains a basis Every spanning list in a vector space can be reduced to a basis of the vector space. Every finite-dimensional vector space has... - answersBasis of finite-dimensional vector space Every finite-dimensional vector space has a basis. Every linearly independent list of vectors in a finite-dimensional vector space can be... - answersLinearly independent list extends to a basis Every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space. Every subspace of V is part of a - answersEvery subspace of V is part of a direct sum equal to V Suppose V is finite-dimensional and U is a subspace of V. Then there is a subspace W of V such that V=U +(direct sum)+ W. Any two bases of a finite-dimensional vector space have the - answersBasis length does not depend on basis Any two bases of a finite-dimensional vector space have the same length. dimension, dim V - answersThe dimension of a finite-dimensional vector space is the length of any basis of the vector space. The dimension of V (if V is finite-dimensional) is denoted by dim V. If V is finite-dimensional and U is a subspace of V - answersDimension of a subspace If V is finite-dimensional and U is a subspace of V, then dim U less than or equal too dim V Suppose V is finite-dimensional. Then every linearly independent list of vectors in V with length dim V... - answersLinearly independent list of the right length is a basis Suppose V is finite-dimensional. Then every linearly independent list of vectors in V with length dim V is a basis of V. Suppose V is finite-dimensional. Then every spanning list of vectors in V with length dim V is a - answersSpanning list of the right length is a basis Suppose V is finite-dimensional. Then every spanning list of vectors in V with length dim V is a basis of V. If U1 and U2 are subspaces of a finite dimensional dim(U1+U2) - answersdim(U1+U2)= dimU1+ dimU2 -dim(U1 intersect U2)