chapter 3 Flashcards
real vector space definition
set V of vectors endowed with operations addition and scalar multiplication, as well as satisfying the 7 vector space axioms
Let V be a vector space and let W be a non-empty subset of V . We say that W is a vector subspace (or simply a subspace) of V if the following holds:
(i) for all w1,w2∈W, we have w1+w2 ∈W and
(ii) for all λ∈R and w∈W, we have λw∈W.
3 vector subspace properties
i) if W is a vector subspace of V , then W contains the null vector
ii) if a subspace W contains a vector w, it also contains its opposite −w, as W is stable under
multiplication by a scalar and −w = (−1).w
iii) a vector subspace W of V is itself a vector space, when endowed with the addition and scalar multiplication coming from V
The subspaces of R2 are the following:
(i) the trivial subspace {0},
(ii) lines through the origin,
(iii) the whole space R2
the set of solutions to a homogeneous system of linear equations of n variables..
forms a vector subspace of Rn.
Let V be a vector space and let u1,…,uk be vectors of V. Then span(u1,…,uk) is..
a vector subspace of V.
A family of vectors u1,…,un is called a basis for V if..
if it is a linearly independent set of vectors that span V .
If a family of vectors v1,v2,…,vn is a basis of a vector space V , then every family of vectors of V containing more than n vectors is
linearly independent
Any two bases for a vector space V contain - vectors.
Any two bases for a vector space V contain the same number of vectors.
Let V be a vector space. We define the dimension of V to be
the number of vectors in any basis of V.
The set of solutions of a homogeneous system of linear equations in n unknowns is
a vector subspace of  n whose dimension is the number of free variables, after reducing the system to its echelon form.
Let V be a vector space, and let W be a subspace. Then
dimW =
dimW=dimV <=>
W=V
