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Section 1.3 Solving Linear Systems (E3)

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Activity 1.3.1.

Free browser-based technologies for mathematical computation are available online.

  • Go to https://sagecell.sagemath.org/.

  • 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 RREF of [132257].

Since the vertical bar in an augmented matrix does not affect row operations, the RREF of [132257] may be computed in the same way.

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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 RREF[231306].
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Activity 1.3.3.

Consider the following system of equations.

3x1−2x2+13x3=62x1−2x2+10x3=2−x1+3x2−6x3=11.
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(a)

Convert this to an augmented matrix and use technology to compute its reduced row echelon form:

RREF[????????????]=[????????????]
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(b)

Use the RREF matrix to write a linear system equivalent to the original system. Then find its solution set.
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Activity 1.3.4.

Consider the vector equation

x1[32−1]+x2[−2−20]+x3[1310−3]=[621]
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(a)

Convert this to an augmented matrix and use technology to compute its reduced row echelon form:

RREF[????????????]=[????????????]
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(b)

Use the RREF matrix to write a linear system equivalent to the original system. Then find its solution set.
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Activity 1.3.5.

Consider the following linear system.

x1+2x2+3x3=12x1+4x2+8x3=0
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(a)

Find its corresponding augmented matrix A and use technology to find RREF(A).

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(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:

x1+2x2=4+x3=−1
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(a)

Let x1=a and write the solution set in the form {[a??]|a∈R}.

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(b)

Let x2=b and write the solution set in the form {[?b?]|b∈R}.

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(c)

Which of these was easier? What features of the RREF matrix [1204001−1] caused this?

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Definition 1.3.7.

Recall that the pivots of a matrix in RREF form are the leading 1s in each non-zero row.

The pivot columns in an augmented matrix correspond to the bound variables in the system of equations (x1,x3 below). The remaining variables are called free variables (x2 below).

[1204001−1]

To efficiently solve a system in RREF form, assign letters to the free variables, and then solve for the bound variables.

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Activity 1.3.8.

Find the solution set for the system

2x1−2x2−6x3+x4−x5=3−x1+x2+3x3−x4+2x5=−3x1−2x2−x3+x4+x5=2

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

2x1−2x2−6x3+x4−x5=3−x1+x2+3x3−x4+2x5=−3x1−2x2−x3+x4+x5=2

may be written as

{[1+5a+2b1+2a+3ba3+3bb]|a,b∈R}.
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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. {[123]}

  • Consistent with infinitely-many solutions: e.g. {[12−3aa]|a∈R}

  • Inconsistent: ∅ or {}

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Subsection 1.3.1 Videos

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Figure 1.3.11. Video: Solving a system of linear equations with infinitely-many solutions
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Exercises 1.3.2 Exercises

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Exercises available at checkit.clontz.org 1 .

https://checkit.clontz.org/#/banks/tbil-la/E3/