Vector Spaces - Examples with Solutions
Introduction to Vector Spaces
Let us consider the following equations:
-
\( 2 x + 3 = 4 \) this equation involves sums of real expressions and multiplications by real numbers
-
\( 2 \lt a , b \gt + 2 \lt 2 , 4 \gt = \lt 7 , 0 \gt \) this equation involves sums of 2-d vectors and multiplications by real numbers
-
\( 2 \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
+ 4 \begin{bmatrix}
1 & 0 \\
-1 & 10
\end{bmatrix}
= \begin{bmatrix}
2 & -3 \\
5 & -6
\end{bmatrix}
\) this equation involves sums of 2 by 2 matrices and multiplications by real numbers
- \( 2 P(x) +3(2 x - 3) = -2(x^2 - 2x - 5) \) this equation involves sums of polynommials and multiplications by real numbers
Classifying sets by their properties helps in solving problems involving different king of mathematical objects such as matrices, polynomials, 2-d vectors, 3-d vectors, n-d vectors, planes in geometry, functions,...and therefore develop ways and methods to solve different problems using the same algorithms.
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:
- The set of all real number \( \mathbb{R} \) associated with the addition and scalar multiplication of real numbers.
- The set of all the complex numbers \( \mathbb{C} \) associated with the addition and scalar multiplication of complex numbers.
- The set of all polynomials \( R_n(x) \) with real coefficients associated with the addition and scalar multiplication of polynomials.
- 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.
- The set of all matrices of dimension \( m \times n \) associated with the addition and scalar multiplication as defined for matrices.
- 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
- Linear Algebra and its Applications - 5 th Edition - David C. Lay , Steven R. Lay , Judi J. McDonald
- Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres
- Matrices with Examples and Questions with Solutions
- Polynomials
- Complex Numbers Add, Subtract and Scalar Multiply Matrices
