Consider the module over
.
Note that is a PID as it
is generated by
.
Also rank of over
is
as
is a basis
of
over
.
Consider the set .
Since so
is a
generating set of
.
But cant be
reduced to a basis.--------------(1)
Again the set is linearly
independent but it cant be extended to a basis because we cant
write every element of
as a linearly
combination of
.---------------------------(2)
Thus (1) and (2) provide examples where the above statements are not true.
ote. lhere Is 1o at the end.) For a vector space over a field, we know...
explain what a basis for a vector space is. How does a basis differ from a span of a vector space? What are some characteristics of a basis? Does a vector space have more than one basis? Be sure to do this: A basis B is a subset of the vector space V. The vectors in B are linearly independent and span V.(Most of you got this.) A spanning set S is a subset of V such that all vectors...
Problem 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...
Give an example of a vector space V finitely generated over a field F , together with nonempty subsets B1, B2, and B3 of V satisfying the following conditions: (1) Each Bi is linearly independent; (2) For each 1≤I ̸= j ≤3 there exists a basis of V containing Bi ∪Bj; (3) There is no basis of V containing B1 ∪ B2 ∪ B3.
Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the set S = {T : V -> F | T is a linear transformation}. Define the operations: (T1 + T2)(v) := T1(v) + T2(v), (aT1)(v) = a(T1(v)) for any v in V, a in F. Prove tat S with these operations is a vector space over F. (b) In S, we have elements fi : V -> F...
Let m, n EN\{1}, V be a vector space over R of dimension n and (v1, ..., Vm) be an m tuple of V. (Select ALL that are TRUE) If m > n then (v1, ..., Vm) spans V. If (v1, ..., Um) is linearly independent then m <n. (v1, ..., Um) is linearly dependent if and only if for all i = 1,..., m we have that U; Espan(vi, .., Vi-1, Vj+1, ..., Um). Assume there exists exactly one...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
Suppose that V is a 3-dimensional vector space over a field F and T : V → V is a linear tion such that the corresponding F[x]-module structure on V is given by 7. V F[x]/(x3-x2-x + 1). Among the matrices A, B, and C given below, which are the matrix of T in some basis for V. Explain 1 1 0 0 0-1 B-10 1 A 0 1 0 0 1 1 0 0 -1 0 0 -1 (Note:...
can anybody explain how to do #9 by using the theorem
2.7?
i know the vectors in those matrices are linearly independent,
span, and are bases, but i do not know how to show them with the
theorem 2.7
a matrix ever, the the col- ons of B. e rela- In Exercises 6-9, use Theorem 2.7 to determine which of the following sets of vectors are linearly independent, which span, and which are bases. 6. In R2t], bi = 1+t...
linear algebra question easy, please answer fast with steps
Mark each statement True or False. Justify each answer. Here A is an mxn matrix. Complete parts (a) through (e) below a. If B is a basis for a subspace H, then each vector in H can be wrben in only one way as a linear combination of the vectors in B. Choose the correct answer below O A. The statement is false. Bases for a subspace H may be linear...