Elementary Linear Algebra Chapter 4: Real Vector Spaces
Welcome!
Note: This page is based on the fourth chapter in Elementary Linear Algebra with Applications (Ninth Edition) by Bernard Kolman and David R Hill.
4.1 Vectors in the Plane and in 3-Space
Basic definitions of what a vector and a coordinate system is (see book). I am under the impression that you have had enough math to know what these are.
4.2 Vector Spaces
A real vector space is a set V of elements on which we have two operations + and ∙ defined with the following properties:
(a) If u and v are any elements in V, then u + v is in V. We say that V is closed under the operation +
1. u + v = v + u for all u, v in V
2. u + (v + w) = (u + v) + w for all u, v, w in V
3. There exists an element 0 in V such that u + 0 = 0 + u = u for any u in V
4. For each u in V there exists an element –u in V such that u + -u = -u + u = 0
(b) If u is any element in V and c is any real number, then c ∙ u is in V
1. c ∙ (u + v) = c ∙ u + c ∙ v for any u, v in V and any real number c
2. (c + d) ∙ u = c ∙ u + d ∙ u for any u in V and any real numbers c and d
3. c ∙ (d ∙ u) = (cd) ∙ u for any u in V and any real numbers c and d
4. 1 ∙ u = u for any u in V
4.3 Subspaces
A subspace is a “mini” vector space that satisfies all of the properties mentioned in section 4.2. An easy way to test if something is a subspace is to see if it satisfies the addition and scalar multiplication properties.
4.4 Span
Let S be a set of vectors in a vector space V. If every vector in V is a linear combination of the vectors in S, then the set S is said to span V, or V is spanned by the set S; that is, span S = V.
4.5 Linear Independence
Linear Independence is when all vectors in a set of vectors are unique. So if there are two vectors in a set that are a combination of other vectors in the the set, then the set is not linear independent.
4.6 Basis and Dimension
The vectors in a vector space V are said to form a basis for V if they:
(1) span V
(2) linear independent
The dimension of a nonzero vector space V is the number of vectors in a basis for V. We often write dim V for the dimension of V. We also define the dimension of the trivial vector space {0} to be zero.
4.7 Homogeneous Systems
4.9 Rank of a Matrix
• If A is an m × n matrix, then rank A + nullity A =n
• A is nonsingular if and only if rank A = n
• If A is an n × n matrix, then rank A = n if and only if det(A) ≠ 0
