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System of Linear Equations
|Definition 1 Fix a set of numbers
, where i = 1, ...,m and
j = 1,... , n. A system of m linear equations in n variables , is
Consistent: It has solutions (one or infinitely
| Solutions to a system of linear equations can be
• Substitution. (Convenient for small systems.)
• Elementary Row Operations . (Convenient for large systems.)
It is not needed to write down the variable while
performing the EROs. Only the coefficients of the system , and the
right hand side is needed.
This is the reason to introduce the matrix notation.
|Elementary Row Operations (EROs)
• Add to one row a multiple of the other.
• Interchange two rows .
• Multiply a row by a nonzero constant.
EROs do not change the solutions of a linear system of equations.
EROs are performed until the matrix is in echelon form.
Echelon form: Solutions of the linear system can be easily read out.
|Definition 2 The diagonal elements of a
are given by ,
for i from 1 to the minimum of m and n.
• Echelon form: Upper triangular.
(Every element below the diagonal is zero .)
• Reduced Echelon Form: A matrix in echelon form such that
the first nonzero element in every row satisfies both,
- it is equal to 1,
- it is the only nonzero element in that column.
|Existence and uniqueness
• A system of linear equations is inconsistent if and only if the
echelon form of the augmented matrix has a row of the form
• A consistent system of linear equations contains either,
- a unique solution, that is, no free variables,
- or infinitely many solutions, that is, at least one free
|Vectors in IRn
• Definition, Operations, Components.
• Linear combinations.
|Definition, Operations, Components
Definition 3 A vector in IRn, n≥1, is an oriented segment.
• Addition, Difference : Parallelogram law .
• Multiplication by a number: Stretching, compressing.
|Some properties of the addition and
multiplication by a scalar:
u + v = v + u,
u + (v + w) = (u + v) + w,
a(u + v) = au + av,
(a + b)u = au + bu.
|Definition 4 A vector w ∈ IRn is a linear
combination of p≥ 1
vectors in IRn if there exist p numbers such
A system of linear equations can be written as a vector equation:
Definition 5 Given p vectors in IRn, denote by
Span the set of all linear combinations of .
• If w ∈ Span , then there exist numbers
|Matrices as linear functions
Definition 6 A linear function y : IRn →IRm is a function y(x)
of the form
and are constants, with i = 1, ,m, and j = 1, n.
|Introducing the vector c ∈ IRm, and the m n
matrix A as follows,
|The product Ax is defined as follows:
Exercise: Show that this product satisfies the
A system of linear equations
can be expressed as a linear function, or as a
linear combination of
|Theorem 1 Fix and m n matrix
b ∈ IRm. Then,
b ∈ Span there exist , such that
the echelon form of [A l b]
has NO row of the form
[0 ...0 l b ≠ 0].