theorems to prove

Theorem 1.4. Any intersection of subspaces of a vector space V is a subspace of V.

Theorem 1.7. Let S be a linearly independent subset of a vector space V , and let v be a vector in
V that is not in S. Then S ∪{v} is linearly dependent if and only if v ∈ span(S).

Theorem 1.10 (Replacement Theorem). Let V be a vector space that is generated by a set G
containing exactly n vectors, and let L be a linearly independent subset of V containing exactly m
vectors. Then m ≤ n and there exists a subset H of G containing exactly n−m vectors such that
L∪H generates V.

Theorem 2.1. Let V and W be vector spaces and T : V → W be linear. Then N(T) and R(T) are subspaces of V and W, respectively

Theorem 2.2. Let V and W be vector spaces, and let T : V → W be linear. If β = {v1,v2,...,vn}
is a basis for V , then R(T) = span(T(β)) = span{T(v1),T(v2),...,T(vn)}.

Theorem 2.3 (Dimension Theorem). Let V and W be vector spaces, and let T : V → W be linear. If V is finite-dimensional, then nullity(T) + rank(T) = dim(V ).

Theorem 2.4. Let V and W be vector spaces, and let T : V → W be linear. Then T is one-to-one
if and only if N(T) ={0}.

Theorem 2.5. Let V and W be finite-dimensional vector spaces of equal dimension, and let T :
V →W be linear. Then the following are equivalent.
(a) T is one-to-one.
(b) T is onto.

Theorem 2.6 (Bread and butter theorem). Let V and W be vector spaces over F, and suppose
that {v1,v2,...,vn} is a basis for V . For w1,w2,...,wn in W, there exists exactly one linear
transformation T : V → W such that
T(vi) = wi for i = 1,2,...,n.

Theorem 2.14. Let V and W be finite-dimensional vector spaces having ordered bases β and γ,
respectively, and let T : V → W be linear. Then, for each u ∈ V, we have
[T(u)]γ = [T]γ
β[u]β.

Theorem 2.19. Let V and W be finite-dimensional vector spaces (over the same field). Then V
is isomorphic to W if and only if dim(V) = dim(W).

Theorem 2.21. For any finite-dimensional vector space V with ordered basis β, φβ is an isomor
phism.

Theorem 3.5. The rank of any matrix equals the maximum number of its linearly independent
columns; that is, the rank of a matrix is the dimension of the subspace generated by its columns.

Theorem 3.7. Let T : V → W and U : W → Z be linear transformations on finite-dimensional
vector spaces V , W, and Z, and let A and B be matrices such that the product AB is defined. Then
(a) rank(UT) ≤ rank(U).
(b) rank(UT) ≤ rank(T).

Theorem 3.9. Let K be the solution set of a consistent system of linear equations Ax = b, and let
KH be the solution set of the corresponding homogeneous system Ax = 0. Then for any solution s
to Ax =b
K ={s}+KH ={s+k:k∈KH}.

Theorem 3.11. Let Ax = b be a system of linear equations. Then the system is consistent if and
only if
rank(A) = rank([A | b]).