Section 1.3 Solving Linear Systems (E3)
Activity 1.3.1.
Free browser-based technologies for mathematical computation are available online.
In the dropdown on the right, you can select a number of different languages. Select "Octave" for the Matlab-compatible syntax used by this text.
Type
rref([1,3,2;2,5,7])
and then press the Evaluate button to compute the of
Since the vertical bar in an augmented matrix does not affect row operations, the
Activity 1.3.2.
In the HTML version of this text, code cells are often embedded for your convenience when RREFs need to be computed.
Try this out to compute
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rref([2,3,1;3,0,6])
Activity 1.3.3.
Consider the following system of equations.
(a)
Convert this to an augmented matrix and use technology to compute its reduced row echelon form:
(b)
Use the
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rref([3,-2,13,6;2,-2,10,2;-1,3,-6,11])
Activity 1.3.4.
Consider the vector equation
(a)
Convert this to an augmented matrix and use technology to compute its reduced row echelon form:
(b)
Use the
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rref([3,-2,13,6;2,-2,10,2;-1,0,-3,1])
Activity 1.3.5.
Consider the following linear system.
(a)
Find its corresponding augmented matrix
(b)
How many solutions do these linear systems have?
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Activity 1.3.6.
Consider the simple linear system equivalent to the system from the previous activity:
(a)
Let
(b)
Let
(c)
Which of these was easier? What features of the RREF matrix
Definition 1.3.7.
Recall that the pivots of a matrix in
The pivot columns in an augmented matrix correspond to the bound variables in the system of equations (
To efficiently solve a system in RREF form, assign letters to the free variables, and then solve for the bound variables.
Activity 1.3.8.
Find the solution set for the system
by row-reducing its augmented matrix, and then assigning letters to the free variables (given by non-pivot columns) and solving for the bound variables (given by pivot columns) in the corresponding linear system.
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Observation 1.3.9.
The solution set to the system
may be written as
Remark 1.3.10.
Don't forget to correctly express the solution set of a linear system. Systems with zero or one solutions may be written by listing their elements, while systems with infinitely-many solutions may be written using set-builder notation.
Consistent with one solution: e.g.
Consistent with infinitely-many solutions: e.g.
Inconsistent:
or
Subsection 1.3.1 Videos
Exercises 1.3.2 Exercises
Exercises available at checkit.clontz.org 1 .https://checkit.clontz.org/#/banks/tbil-la/E3/