Introduction to Vector Spaces

Let us consider the following equations:

1.     \( 2 x + 3 = 4 \)      this equation involves sums of real expressions and multiplications by real numbers
   
2.     \( 2 \lt a , b \gt + 2 \lt 2 , 4 \gt = \lt 7 , 0 \gt \)      this equation involves sums of 2-d vectors and multiplications by real numbers
   
3.     \( 2 \begin{bmatrix} a & b \\ c & d \end{bmatrix} + 4 \begin{bmatrix} 1 & 0 \\ -1 & 10 \end{bmatrix} = \begin{bmatrix} 2 & -3 \\ 5 & -6 \end{bmatrix} \)      this equation involves sums of 2 by 2 matrices and multiplications by real numbers
   
4.     \( 2 P(x) +3(2 x - 3) = -2(x^2 - 2x - 5) \)      this equation involves sums of polynommials and multiplications by real numbers

Definition of a Vector Space

In what follows, vector spaces (1 , 2) are in capital letters and their elements (called vectors) are in bold lower case letters.
A nonempty set \( V\) whose vectors (or elements) may be combined using the operations of addition (+) and multiplication (\( \cdot \) ) by a scalar is called a vector space if the conditions in A and B below are satified:
Note An element or object of a vector space is called vector.
A)     the addition of any two vectors of \( V\) and the multiplication of any vectors of \( V\) by a scalar produce an element that belongs to \( V\).
       Let \( \textbf{u}\) and \( \textbf{v} \) be any two elelments of the set \( V\) and \( r \) any real number.
       1) \( \textbf{u} + \textbf{v} = \textbf{w}\)   ,   \( \textbf{w} \) is an element of the set \( V\) ; we say the set \( V\) is closed under vector addition
       2) \( r \cdot \textbf{u} = \textbf{z} \)   ,   \( \textbf{z} \) is an element of the set \( V\) we say the set \( V\) is closed under scalar multiplication

B)     For any vectors \( \textbf{u}, \textbf{v}, \textbf{w} \) in \( V\) and any real numbers \( r \) and \( s \), the two operations described above must obey the following rules :
       3) Commutatitivity of vector addition :        \( \textbf{u} + \textbf{v} = \textbf{v} + \textbf{u} \)
       4) Associativity of vector addition :        \( (\textbf{u} + \textbf{v}) +\textbf{w} = \textbf{v} + ( \textbf{u} + \textbf{w}) \)
       5) Associativity of multiplication:        \( r \cdot (s \cdot \textbf{u}) = (r \cdot s) \cdot \textbf{u} \)
       6) A zero vector \( \textbf{0} \) exists in \( \textbf{v}\) and is such that for any element \( \textbf{u}\) in the set \( \textbf{v}\), we have: \( \textbf{u} + \textbf{0} = \textbf{u} \)
       7) For each vector \( \textbf{u}\) in \( V\) there exists a vector \( - \textbf{u} \) in \( V\), called the negative of \( \textbf{u}\), such that: \( \textbf{u} + (- \textbf{u}) = \textbf{0} \)
       8) Distributivity of Addition of Vectors:        \( r \cdot (\textbf{u} + \textbf{v} ) = r \cdot \textbf{u} + r \cdot \textbf{v} \)
       9) Distributivity of Addition of Real Numbers:        \( (r + s) \cdot \textbf{u} = r \cdot \textbf{u} + s \cdot \textbf{u} \)
       10) For any element \( \textbf{u}\) in \( V\) we have:        \( 1 \cdot \textbf{u} = \textbf{u} \)

NOTES
1) Although the element of a vector space is called vector, a vector space may be a set of matrices, functions, solutions to differential equations, 3-d vectors, ....,They do not have to be VECTORS of n dimensions such as 2 or 3 dimensional vectors used in physics.

Examples of Vector Spaces

Example 1 The following are examples of vector spaces:

1. The set of all real number \( \mathbb{R} \) associated with the addition and scalar multiplication of real numbers.
2. The set of all the complex numbers \( \mathbb{C} \) associated with the addition and scalar multiplication of complex numbers.
3. The set of all polynomials \( R_n(x) \) with real coefficients associated with the addition and scalar multiplication of polynomials.
4. The set of all vectors of dimension \( n \) written as \( \mathbb{R}^n \) associated with the addition and scalar multiplication as defined for 3-d and 2-d vectors for example.
5. The set of all matrices of dimension \( m \times n \) associated with the addition and scalar multiplication as defined for matrices.
6. The set of all functions \( \textbf{f} \) satisfying the differential equation \( \textbf{f} = \textbf{f '} \)

Example 2
Proove that the set of all 2 by 2 matrices associated with the matrix addition and the scalar multiplication of matrices is a vector space.
Solution to Example 2
Let \( V\) be the set of all 2 by 2 matrices.
1) Addition of matrices gives
\( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} = \begin{bmatrix} a+a' & b+b' \\ c+c' & d+d' \end{bmatrix} \)
Adding any 2 by 2 matrices gives a 2 by 2 matrix and therefore the result of the addition belongs to \( V\).

2) Scalar multiplication of matrices gives gives
\( r \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} r a & r b \\ r c & r d \end{bmatrix} \)
Multiply any 2 by 2 matrix by a scalar and the result is a 2 by 2 matrix is an element of \( V\).

3) Commutativity
\( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} \\\\ = \begin{bmatrix} a+a' & b+b' \\ c+c' & d+d' \end{bmatrix} \\\\ = \begin{bmatrix} a'+a & b'+b \\ c'+c & d'+d \end{bmatrix} \\\\ = \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} + \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

4) Associativity of vector addition
\( \left ( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} \right) + \begin{bmatrix} a'' & b'' \\ c'' & d'' \end{bmatrix} \\ = \begin{bmatrix} a+a' & b+b' \\ c+c' & d+d' \end{bmatrix} + \begin{bmatrix} a'' & b'' \\ c'' & d'' \end{bmatrix} \\\\ = \begin{bmatrix} (a+a')+a'' & (b+b')+b'' \\ (c+c') + c''& (d+d')+d'' \end{bmatrix} \\\\ = \begin{bmatrix} a+(a'+a'') & b+(b'+b'') \\ c+(c' + c'')& d+(d'+d'') \end{bmatrix} \\\\ = \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \left( \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} + \begin{bmatrix} a'' & b'' \\ c'' & d'' \end{bmatrix} \right) \)

5) Associativity of multiplication
\( r \left( s \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = r \left( \begin{bmatrix} s a & s b \\ s c & s d \end{bmatrix} \right) \\\\ = \begin{bmatrix} r s a & r s b \\ r s c & r s d \end{bmatrix} \\\\ = \begin{bmatrix} (r s) a & (r s) b \\ (r s) c & (r s) d \end{bmatrix} \\\\ = (r s) \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

6) Zero vector
\( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \\\\ = \begin{bmatrix} a+0 & b+0 \\ c+0 & d+0 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

7) Negative vector
\( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} - a & - b \\ - c & - d \end{bmatrix} \\\\ = \begin{bmatrix} a+(-a) & b+(-b) \\ c+(-c) & d+(-d) \end{bmatrix} \\\\ = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

8) Distributivity of sums of matrices:
\( r \left ( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} \right) \\\\ = \begin{bmatrix} r(a+a') & r(b+b') \\ r(c+c') & r(d+d') \end{bmatrix} \\\\ = \begin{bmatrix} r a+ r a' & r b+ r b \\ r c+r c' & r d+ r d \end{bmatrix} \\\\ = r \left ( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) + r \left(\begin{bmatrix} a' & b' \\ c' & d' \end{bmatrix} \right) \)

9) Distributivity of sums of real numbers:
\( (r + s ) \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} (r + s ) a & (r + s ) b \\ (r + s ) c & (r + s ) d \end{bmatrix} \\\\= \begin{bmatrix} r a + s a & r b + s b \\ r c + s c & r d + s d \end{bmatrix} \\\\= \begin{bmatrix} r a & r b \\ r c & r d \end{bmatrix} + \begin{bmatrix} s a & s b \\ s c & s d \end{bmatrix} \\\\= r \begin{bmatrix} a & b \\ c & d \end{bmatrix} + s \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

10) Multiplication by 1.
\( 1 \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 a & 1 b \\ 1 c & 1 d \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

Example 3
Show that the set of all real functions continuous on \( (-\infty,\infty) \) associated with the addition of functions and the multiplication of matrices by a scalar form a vector space.
Solution to Example 3
From calculus, we know if \( \textbf{f} \) and \( \textbf{g} \) are real continuous functions on \( (-\infty,\infty) \) and \( r \) is a real number then
\( (\textbf{f} + \textbf{g})(x) = \textbf{f}(x) + \textbf{g}(x) \) is also continuous on \( (-\infty,\infty) \)
and
\( r \textbf{f}(x) \) is also continuous on \( (-\infty,\infty) \)
Hence the set of functions continuous on \( (-\infty,\infty) \) is closed under addition and scalar multiplication (the first two conditions above).
The remaining 8 rules are automatically satisfied since the functions are real functions.

Example 4
Show that the set of all real polynomials with a degree \( n \le 3 \) associated with the addition of polynomials and the multiplication of polynomials by a scalar form a vector space.
Solution to Example 4
The addition of two polynomials of degree less than or equal to 3 is a polynomial of degree lass than or equal to 3.
The multiplication, of a polynomial of degree less than or equal to 3, by a real number results in a polynomial of degree less than or equal to 3
Hence the set of polynomials of degree less than or equal to 3 is closed under addition and scalar multiplication (the first two conditions above).
The remaining 8 rules are automatically satisfied since the polynomials are real.

Example 5 Show that the set of polynomials with a degree \( n = 4 \) associated with the addition of polynomials and the multiplication of polynomials by a real number IS NOT a vector space.
Solution to Example 5
The addition of two polynomials of degree 4 may not result in a polynomial of degree 4.
Example: Let \( \textbf{P}(x) = -2 x^4+3x^2- 2x + 6 \) and \( \textbf{Q}(x) = 2 x^4 - 5x^2 + 10 \)
\( \textbf{P}(x) + \textbf{Q}(x) = (-2 x^4+3x^2- 2x + 6 ) + ( 2 x^4 - 5x^2 + 10) = - 5x^2 - 2 x + 16 \)
The result is not a polynomial of degree 4. Hence the set is not closed under addition and therefore is NOT vector space.

Example 6
Show that the set of integers associated with addition and multiplication by a real number IS NOT a vector space
Solution to Example 6
The multiplication of an integer by a real number may not be an integer.
Example: Let \( x = - 2 \)
If you multiply \( x \) by the real number \( \sqrt 3 \) the result is NOT an integer.

More References and links

1. Linear Algebra and its Applications - 5 th Edition - David C. Lay , Steven R. Lay , Judi J. McDonald
2. Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres
3. Matrices with Examples and Questions with Solutions
4. Polynomials
5. Complex Numbers Add, Subtract and Scalar Multiply Matrices