Introduction to Linear Systems

Linear Algebra

System of Equations, Augmented Matrices, Row operations, Gaussian Elimination

Multiple Choice

An m x n system of linear equations consists of m linear equations in n unknowns. Describe the system to the left.

• 3 x 3

• 2 x 2

• 2 x 3

• 3 x 2

An m x n system linear equation is:

- inconsistent if it has no solution

- consistent if it has at least one solution

Multiple Choice

What can you say about the system?

$$x\ +\ 2y\ =\ 3$$

$$x\ +\ 2y\ =\ 4$$

• Consistent

• Inconsistent

Multiple Choice

What can you say about the system?

$$x\ +\ 3y\ =\ 5$$

$$2x\ +\ 6y\ =\ 10$$

• Consistent

• Inconsistent

Multiple Select

A consistent system can have:

• No solution

• One solution

• Infinitely many solutions

If a consistent system has exactly one solution, it is independent. If a consistent system has infinitely many solutions it is dependent.

Match

Match the following:

Inconsistent

Consistent and independent

Consistent and dependent

• Inconsistent

• Consistent and independent

• Consistent and dependent

An m x n system linear equations is:

• Underspecified if it has more unknowns that equations (n > m)

• Overspecified if it has more equations than unknowns (m > n)

Draw

How is it going so far?

Tell me via a sketch

A system of m x n linear equations can be written as a rectangular array as seen in the photo to the right.

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Augmented Matrices

Multiple Choice

Write the augmented matrix that corresponds to the system

• A

• B

• C

• D

Multiple Choice

Which of the following is the correct representation for the system of equations?

• A

• B

• C

• D

Multiple Select

What is left out in an augmented matrix?

• + sign

• = sign

• variables

• coefficients

Multiple Choice

What is the coefficient of x₃ from the second equation?

• -1

• 1

• 2

• 7

Multiple Choice

Which of the following represents the third equation?

• $$9x_1+x_{2\ }=1$$

• $$9x_{1\ }+2x_{2\ }=1$$

• $$9x_{1\ }+2x_{3\ }=1$$

• $$9x_{1\ }-2x_{3\ }=1$$

Multiple Choice

Write the system that corresponds to the augmented matrix.

• A

• B

• C

• D

Multiple Choice

Write the system of linear equations represented by the augmented matrix

• First option

• Second option

• Third option

• Fourth option

Multiple Choice

Write the system of linear equations and solve.

• (2,7,5)

• (-8,7,5)

• (-8,2,5)

• (-3,2,5)

Open Ended

What was special in that augmented matrix? Why was it so easy to solve the system of equations?

Multiple Choice

Back-substitute to calculate the solution from the augmented matrix Matrix.

• 2, 4, (1/2)

• (-1/5), 4, (1/2)

• (1/2), 2, 1

• (1/5) , 4, (1/2)

Multiple Choice

Solve the system of equations represented by the augmented matrix.

• (52, 4, 36)

• (4, 1, 6)

• (-53, 4, -36)

• (1, 4, 6)

Multiple Choice

What is the solution to the systems of equations represented by the 3x4 Matrix?

• (4, 6, 2)

• (-6, 2, 2)

• All Real Numbers

• No Solutions

A rectangular matrix is in row echelon form if it has the following three properties:

• All nonzero rows are above any rows of all zeros

• Each leading entry of a row is in a column to the right of the leading entry of the row above it

• All entries of a column below a leading entry are zeros
  

Very practical, but how to get it?

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Row operations

From our previous study of simultaneous equations, we know that we can do the following things without changing the solutions:

Some text here about the topic of discussion

1. Interchange Equations

Some text here about the topic of discussion

2. Replace an equation by any non-zero multiple of itself

Some text here about the topic of discussion

3. Replace an euation by a multiple of itself plus a multiple of another equation

Some text here about the topic of discussion

Drag and Drop

These principles can also be applied to augmented matrices as ​

Placeholder

​

We can hence:





1. ​

Placeholder

rows

2. Replace any row by a non-zero ​

Placeholder

of itself

3. Replace any row by ​

Placeholder

​

Placeholder

Drag these tiles and drop them in the correct blank above

row operations

Interchange

multiple

itself

plus a multiple of another row

Match

Match the following

R₁ R₂

-3R₁ R₁

2R₂ - R₁ R₂

• R₁ $$\leftrightarrow$$ R₂

• -3R₁ $$\rightarrow$$ R₁

• 2R₂ - R₁ $$\rightarrow$$ R₂

Multiple Choice

Complete the row operation.

$$R_2=-2r_1+r_2$$  

• 1 -3 |-2

• 0  1 | 2

• -2  5 |  5

• 1  0  | 8

Open Ended

On paper, write down the new augmented matrix after row reduction and observe it. In your opinion what are row reductions useful for?

Extension

Multiple Choice

Write the system as an augmented matrix and multiply the first row by 3.

$$x+y=4$$

$$3x+3y\ =\ a$$

For what value(s) of a is the system consistent?

• 3

• 4

• 12

Multiple Choice

Write the system as an augmented matrix and multiply the second row by -2.

$$2x-y=5$$

$$-x+ay=3$$

For what value(s) of a is the system inconsistent?

• -1

• 1

• -0.5

• 0.5

Multiple Choice

• A

• B

• C

• D

Open Ended

Write the system as an augmented matrix and replace the second row with "the sum of the first two rows".

$$x+y+z=p$$

$$x\ \ \ \ \ \ +2z=q$$

$$2x+y+3z=r$$

Find the relationship between p, q, and r such that the system is consistent.

Poll

Does linear algebra look interesting so far?

• YES!

• Meh

• No