8.3 Systems of Equations and Matrices: Introduction to Notation
In this section, you will practise reading out mathematical symbols. This is important because your lecturers will often read mathematical statements in lectures, and you will also need to communicate about mathematics with others.
Mathematics uses two kinds of systems of communication:
- Words (for example, 'plus', 'minus', 'equals').
- Mathematical symbols (for example, [latex]+[/latex], [latex]-[/latex], [latex]=[/latex]).
The same mathematical symbols are used in every language, but you still need to know how to say them in English. Learning to read out symbols will help you understand your teachers, communicate mathematics with others, and read mathematical texts more easily.
A linear system of equations is a collection of one or more linear equations in the same unknowns. In a linear equation, each unknown appears only to the first power, variables are not multiplied together, and the unknowns occur only in terms of the form coefficient times variable. The general form of a linear system of [latex]m[/latex] equations in [latex]n[/latex] unknowns [latex]x_1, x_2, \ldots, x_n[/latex] is:
[latex]a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1[/latex]
[latex]a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2[/latex]
[latex]\vdots[/latex]
[latex]a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m.[/latex]
The first equation, for example, is read out as: a one one x one plus, a one two x two plus, dot dot dot plus, a one n x n, is equal to b one. The rest are read out similarly.
The quantities [latex]a_{ij}[/latex] (a i j) are called the coefficients, and the quantities [latex]b_1, b_2, \ldots, b_m[/latex] (b one, b two, dot dot dot, b m) form the right-hand side of the system. The quantities [latex]x_1, x_2, \ldots, x_n[/latex] (x one, x two, dot dot dot, x n) are the unknowns or variables.
Linear systems of equations are usually represented in matrix form:
[latex]Ax = b,[/latex]
where [latex]A[/latex] is a matrix of size [latex]m \times n[/latex] (m by n), [latex]x[/latex] is an [latex]n[/latex]-dimensional vector representing the unknowns, and [latex]b[/latex] is an [latex]m[/latex]-dimensional vector representing the constants on the right-hand side.
This is read out as: A x is equal to b.
A matrix (the plural is matrices) is a rectangular array of numbers arranged in rows and columns. Each number in the matrix is called an entry. The entry in row [latex]i[/latex] and column [latex]j[/latex] is denoted [latex]a_{ij}[/latex] (a i j). A matrix with [latex]m[/latex] rows and [latex]n[/latex] columns is said to have size [latex]m \times n[/latex] (m by n). Note that the number of rows is always stated first. A matrix [latex]A[/latex] is written as:
[latex]A = (a_{ij})_{m \times n},[/latex]
which is read out as A is equal to the m by n matrix with entries a i j.
For example, the general [latex]2 \times 2[/latex] matrix is
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}.\]
Note that matrices are read top to bottom, left to right. Hence, this is read as the two by two matrix a, b, c, d.
The determinant of the [latex]2 \times 2[/latex] matrix given above is:
\[\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc.\]
This is read out as: the determinant of the two by two matrix a, b, c, d equals, a d minus b c.
A convenient way to represent a linear system is as an augmented matrix. The augmented matrix of the system [latex]Ax = b[/latex] is formed by placing the entries of [latex]A[/latex] alongside the entries of [latex]b[/latex], separated by a vertical bar:
\[
\left(
\begin{array}{cccc|c}
a_{11} & a_{12} & \cdots & a_{1n} & b_{1} \\
a_{21} & a_{22} & \cdots & a_{2n} & b_{2} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} & b_{m}
\end{array}
\right).
\]
This is read out as: the augmented matrix with entries a i j, of size m by n, with right-hand side b.
The vertical bar separates the coefficients of the unknowns from the right-hand side of the system.
Language Focus 1
Several important terms in the topic of Systems of Equations and Matrices also have familiar general meanings in everyday English. It is important to distinguish between the two.
The word system in general English means a set of connected things or parts forming a complex whole, for example, the transport system. In mathematics, a system of equations refers specifically to a set of equations that are to be solved simultaneously. The meanings are similar (a collection of related things) but its mathematical use is more precise.
The word solution in general English means an answer to a problem or a way of dealing with a difficult situation. In mathematics, a solution of a system of equations is a specific set of values of the unknowns that satisfies all the equations simultaneously. Note that a system may have no solution, exactly one solution, or infinitely many solutions; an important distinction that does not arise in the everyday use of the word. A system that has at least one solution is called consistent, whereas a system that has no solutions is called inconsistent.
The word free in general English means not restricted or not costing anything. In mathematics, a free variable is an unknown corresponding to a non-pivot column in the coefficient matrix after row reduction, and may therefore take any value. The mathematical use captures the idea of being unrestricted, but in a precise, technical sense.
You may wish to keep a list of mathematical terms that also have a general, non-technical meaning. Further examples from this topic include independent, identity, and pivot.
Gaussian elimination (named after Carl Friedrich Gauss), also called row reduction, is a method for solving a linear system by applying a sequence of row operations to the augmented matrix. There are three types of elementary row operation:
-
Multiply any row by a non-zero number.
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Add any multiple of one row to another row.
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Interchange any two rows.
The natural end stage of Gaussian elimination is reduced row echelon form, abbreviated r.r.e.f. A matrix is in reduced row echelon form if:
-
Any zero rows appear at the bottom of the matrix.
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The first non-zero entry in each non-zero row (called the pivot) is equal to [latex]1[/latex].
-
Each pivot appears strictly to the right of the pivot in the row above.
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Each pivot is the only non-zero entry in its column.
The notation [latex]\text{rref}(A)[/latex] (the row reduced echelon form of A) is sometimes used to denote the reduced row echelon form of a matrix [latex]A[/latex].
Matrix multiplication of a matrix [latex]A[/latex] of size [latex]L \times M[/latex] (L by M) and a matrix [latex]B[/latex] of size [latex]M \times N[/latex] (M by N) gives a new matrix [latex]C[/latex] of size [latex]L \times N[/latex] (L by N), denoted:
[latex]C = AB,[/latex]
The individual entries of [latex]C[/latex] are defined by:
[latex]c_{ij} = \sum_{k=1}^{M} a_{ik} b_{kj}.[/latex]
Note that matrix multiplication is only defined when the number of columns of [latex]A[/latex] equals the number of rows of [latex]B[/latex]. Each entry [latex]c_{ij}[/latex] is the dot product of row [latex]i[/latex] of [latex]A[/latex] and column [latex]j[/latex] of [latex]B[/latex]. For more details about the dot product, see Chapter 6: Vectors and Coordinate Systems.
Mathematics Focus 1: Properties of matrix multiplication
An important property of matrix multiplication is that it is non-commutative, since in general:
[latex]AB \neq BA[/latex].
This means that the order that we multiply matrices matters. For more details about commutativity, see the Mathematics Focus in Section 8.5.
However, the associative property of matrix multiplication does hold:
[latex]A(BC) = (AB)C[/latex],
for three matrices [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex] whose sizes are compatible for multiplication.
Furthermore, matrix multiplication also satisfies the distributive property:
[latex]A(B + C) = AB + AC[/latex]
Note that this is analogous to the distributive property of ordinary multiplication over addition.
A square matrix is a matrix in which the number of rows equals the number of columns, that is, a matrix of size [latex]n \times n[/latex]. The name square reflects the shape of the array: since the number of rows equals the number of columns, the entries form a square arrangement, as opposed to a rectangular arrangement in the general case. For a square matrix, the main diagonal (or principal diagonal) is the diagonal line of entries from the top-left entry to the bottom-right entry. Furthermore, a diagonal matrix is a matrix in which entries outside the main diagonal are [latex]0[/latex].
The identity matrix [latex]I[/latex] is the square matrix in which every entry along the main diagonal is equal to [latex]1[/latex] and every other entry is [latex]0[/latex]. Multiplying any matrix by the identity matrix leaves it unchanged:
[latex]AI = IA = A.[/latex]
This is read out as: A times I equals, I times A, which is equal to A.
The inverse matrix of a square matrix [latex]A[/latex] is the matrix denoted by [latex]A^{-1}[/latex] (A inverse), such that:
[latex]AA^{-1} = A^{-1}A = I.[/latex]
This is read out as: A times A inverse equals, A inverse times A, equals I.
Not every square matrix has an inverse. A square matrix that has an inverse is called invertible or non-singular; one that does not is called non-invertible or singular.
The transpose of a matrix [latex]A[/latex], denoted [latex]A^T[/latex] (A transpose), is the matrix obtained by exchanging the rows and columns of [latex]A[/latex]. That is, the entry in row [latex]i[/latex] and column [latex]j[/latex] of [latex]A^T[/latex] is the entry in row [latex]j[/latex] and column [latex]i[/latex] of [latex]A[/latex]:
[latex]A^T_{ij} = A_{ji}.[/latex]
This is read out as: the i j entry of A transpose equals the j i entry of A.
Mathematics Focus 2: Linear transformations
Matrices can be used to represent linear transformations, that is, operations that map points in a plane or in space to new positions while preserving vector addition and scalar multiplication, and mapping the origin to itself. Two important examples of linear transformations are reflections and rotations. A third important transformation, translation, is closely related but is not a linear transformation in the strict sense, since it does not preserve the origin. This is discussed separately below.
A rotation by an angle [latex]\theta[/latex] (theta) anticlockwise about the origin in two dimensions is represented by the matrix:
\[R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}.\]
Note that a clockwise rotation by [latex]\theta[/latex] is equivalent to an anticlockwise rotation by [latex]-\theta[/latex], which is why the matrix is denoted [latex]R(-\theta)[/latex].
Coordinate rotations in three dimensions can also be represented in matrix form, using [latex]3 \times 3[/latex] (three by three) rotation matrices.
A reflection across the [latex]x[/latex]-axis (x-axis) is represented by:
\[M_x = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.\]
A reflection across the [latex]y[/latex]-axis is represented by:
\[M_y = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}.\]
A translation shifts every point by a fixed amount in a given direction. In two dimensions, a translation by a vector [latex]\begin{pmatrix} a \\ b \end{pmatrix}[/latex] maps a point [latex]\begin{pmatrix} x \\ y \end{pmatrix}[/latex] to the point [latex]\begin{pmatrix} x + a \\ y + b \end{pmatrix}[/latex].
A translation is not a linear transformation in the strict sense, since it does not map the origin to itself, unless [latex]a = b = 0[/latex], in which case it is the identity transformation.
Language Focus 2: The language of transformations
The three transformation types introduced above each have a distinct set of associated language. Note the following patterns:
Rotation:
-
a rotation of [latex]\theta[/latex] radians anticlockwise about the origin
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a rotation of [latex]\theta[/latex] radians clockwise about the origin
-
to rotate a point through [latex]\theta[/latex] radians
-
Note: British English uses anticlockwise, whereas American English uses counterclockwise.
Reflection:
-
a reflection across the [latex]x[/latex]-axis / [latex]y[/latex]-axis
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a reflection about the line [latex]y = x[/latex]
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to reflect a point across a line
-
Note: both across and about are used with reflection in mathematical English; both are correct.
Translation:
-
a translation by the vector [latex]\begin{pmatrix} a \\ b \end{pmatrix}[/latex]
-
to translate a point by [latex]a[/latex] units in the [latex]x[/latex]-direction and [latex]b[/latex] units in the [latex]y[/latex]-direction
These expressions follow the pre-modification patterns discussed in the Language Focus below. Note in particular: rotation of theta degrees anticlockwise (Noun + of + Noun phrase) and coordinate rotation (Noun + Noun).
The mathematical language of this chapter makes extensive use of pre-modification, that is, placing a word or phrase before a noun to qualify or specify its meaning. There are four standard patterns:
- Adjective + Noun
augmented matrix, reduced form, square matrix, free variable, arbitrary parameter, linear transformation - Noun + Noun
row operation, solution set, matrix multiplication, pivot column, identity matrix, coordinate rotation - Proper Adjective + Noun
Gaussian elimination, Cartesian coordinates - Noun + Preposition + Noun phrase
rotation about the origin, reflection across the x-axis, translation by a vector, system of equations, method of elimination, number of rows
For more practice on pre-modifying nouns, see Language Focus 3 in Section 5.3.
In the next section, you will learn how to combine mathematical symbols to make complete mathematical statements, instead of studying symbols separately as you did in this section.
