1.1 Systems of Linear Equations Flashcards
In two dimensions a line in a rectangular xy-coordinate system can be represented by an equation of the form…
ax + by = c
a, b not both 0
In three dimensions a plane in a rectangular xyz-coordinate system can be represented by an equation of the form…
ax + by + cz = d
a, b, c not all 0
We say that ax + by = c is a _ equation in _.
linear equation in the variables x and y.
We say that ax + by + cz = d is a _ equation in _.
linear equation in the variables x, y and z.
Define a linear equation.
We define a linear equation in the n variables x1, x2, … , xn to be one that can be expressed in the form:
a1x1 + a2x2 + … + anxn = b
where a1, a2… an and b are constants and not all a’s are zero.
What is a homogenous linear equation?
When its constant part is zero. i.e. where:
a1x1 + a2x2 + … + anxn = 0
What do we call a linear equation where a1x1 + a2x2 + … + anxn = 0 ?
A homogeneous linear equation.
What do we call a linear equation where its constant part is zero?
Homogeneous
true / false
A linear equation can include products or roots of variables.
False.
A linear equation does not involve any products or roots of variables.
true / false
A linear equation does not involve any products or roots of variables.
True
true / false
In a linear equation all variables occur only to the first power.
True
true / false
In a linear equation, variables may be raised to a power
False
In a linear equation all variables occur only to the first power.
true / false
In a linear equation, variables may be arguments of trigonometric, logarithmic or exponential functions.
False
true / false
In a linear equation, variables do NOT appear as arguments of trigonometric, logarithmic or exponential functions.
True
A finite set of linear equations is called…
A system of linear equations, or more briefly,
a linear system
What is a system of linear equations.
aka a ‘linear system’,
is a finite set of linear equations
If a linear system has no solutions we say it is…
inconsistent
If we say that a linear system is consistent, we mean…
that it has at least one solution
Describe, in geometric terms, the possible solutions of a linear system of two unknowns.
The lines may be parallel and distinct (no intersection and no solution).
The lines may intersect at one point (exactly one solution).
The lines may coincide (infinitely many solutions).
Every system of linear equations has … solutions.
either zero, one, or infinitely many solutions.
A solution to a linear system in n unknowns is…
A sequence s1, s2 … sn where the substitution x1 = s1, x2 = s2 etc makes each and every equation in the system a true statement.
Solutions are usually written as: (s1, s2, … , sn) which is called an ordered n-tuple.
We say that a linear system is consistent if…
…if it has at least one solution.
We say that a linear system is inconsistent if…
…if it has no solutions.
If we know a linear system to have at least one solution we say that the system is…
consistent
If we know a linear system to have no solution we say that the system is…
inconsistent
Every system of linear equations has how many solutions?
Zero, one, or infinitely many solutions. There are no other possibilities.
A consistent linear system has how many solutions?
Either exactly one or infinitely many.
What are the elementary row operations?
- Multiply a row by a nonzero constant.
- Interchange two rows.
- Add a constant times one row to another.
We say that two systems of equations are equivalent if…
…if they have the same solution set.
If two systems of equations have the same solution set, we say that the systems are…
…are equivalent.
