#12 a) We will not have a spanning set of V because one of two things can happen: i) x(k+1) is linearly dependent of the spanning set or ii) x(k+1) is linearly independent.
If it is i) then by definition, x(k+1) is not a member of the spanning set of V (all members must be linearly independent of each other).
On the other ...
Solution Summary
These solves several problems regarding the addition and deletion of vectors from spanning sets.
... so we can have an infinite number of spanning sets that span the same ...spanning vectors to the spanning set, we get a new spanning set that spans the same ...
... independent sets of columns; c) list all minimal sets of columns which span all columns; a',b ... Let's denote the column vectors of by , , , , and the row ...
... Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication. ...
... Thm 5.2.4: If S = {v1, v2, ..., vr}and S' = {w1, w2, ..., wk} are two sets of vectors in a vector space V, then. span{v1, v2, ..., vr}= span{w1, w2, ..., wk}. ...
... Suppose that U and W are finitely-generated subspaces of a vector space V. Prove that U + W is also finitely-generated by finding an explicit spanning set of U ...
... and the number of vectors of the basis which span a given ... according to (17), the subspace W will be spanned by the ... of V, there will be an unique set of values ...
... c) Let S be a subspace in Rn and {x1, x2, ..., xn} be a set of vectors whose span is S. The only vector d such that the cross product of d and xi is 0 for all ...
... 1. Consider the set of vectors: V1 = [6, 0, 2]. V2 = [2, -1, 1], V3 = [-4, -1, -1]. ... What is the dimension of space spanned by these vectors? ...
... consider the span of the vectors that have been added ... This shows how to extend a linearly independent set to obtain ... 1) Consider the given vectors as row matrices ...
... v 3 vp is an orthogonal basis of W and has the same span as. ... 2 Let v1, v2 , v 3 vp be the basis that spans W ... If we set a matrix. ...
