Elementary Linear Algebra Chapter 4: Real Vector Spaces

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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

      The operation "+" is called vector addition and the operation "∙" is scalar multiplication.  
      

      Examples:

1.  Let Rⁿ be the set of all 1 x n matrices [a₁ a₂ ··· a_n], where we define "+" by

                               [a₁ a₂ ··· a_n] + [b₁ b₂ ··· b_n] = [a₁+ b₁   a₂ + b₂ ··· a_n + b_n] and we define "·" by c · [a₁ a₂ ··· a_n] = [ca₁ ca₂ ··· ca_n]

                2.   A polynomial (in t) is a function that is expressible as: 

                               p(t) = a_ntⁿ + a_n-1t^n-1 + ··· + a₁t + a₀

                               q(t) = b_ntⁿ + b_n-1t^n-1 + ··· + b₁t + b₀

                      where a and b are real numbers and n is a nonnegative integer. If a_n ≠ 0, then the function is of degree n. 
                                             We define p(t) + q(t) as: 

                                                        p(t) + q(t) = (a_n+b_n)tⁿ + (a_n-1 + b_n-1)t^n-1 + ··· + (a₁ + b₁)t + (a₀ + b₀)

                                              If c is a scalar, we define c · p(t) as:

                                                        c · p(t) = (ca_n)tⁿ + (ca_n-1)t^n-1 + ··· + (ca₁)t + (ca₀)

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

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