# Precalculus Examples

Step 1

Write as an augmented matrix for .

Step 2

Step 2.1

Multiply each element of by to make the entry at a .

Step 2.1.1

Multiply each element of by to make the entry at a .

Step 2.1.2

Simplify .

Step 2.2

Perform the row operation to make the entry at a .

Step 2.2.1

Perform the row operation to make the entry at a .

Step 2.2.2

Simplify .

Step 2.3

Perform the row operation to make the entry at a .

Step 2.3.1

Perform the row operation to make the entry at a .

Step 2.3.2

Simplify .

Step 2.4

Multiply each element of by to make the entry at a .

Step 2.4.1

Multiply each element of by to make the entry at a .

Step 2.4.2

Simplify .

Step 2.5

Perform the row operation to make the entry at a .

Step 2.5.1

Perform the row operation to make the entry at a .

Step 2.5.2

Simplify .

Step 2.6

Perform the row operation to make the entry at a .

Step 2.6.1

Perform the row operation to make the entry at a .

Step 2.6.2

Simplify .

Step 3

Use the result matrix to declare the final solution to the system of equations.

Step 4

Write a solution vector by solving in terms of the free variables in each row.

Step 5

Write the solution as a linear combination of vectors.

Step 6

Write as a solution set.

Step 7

The solution is the set of vectors created from the free variables of the system.

Step 8

Step 8.1

List the vectors.

Step 8.2

Write the vectors as a matrix.

Step 8.3

To determine if the columns in the matrix are linearly dependent, determine if the equation has any non-trivial solutions.

Step 8.4

Write as an augmented matrix for .

Step 8.5

Find the reduced row echelon form.

Step 8.5.1

Multiply each element of by to make the entry at a .

Step 8.5.1.1

Multiply each element of by to make the entry at a .

Step 8.5.1.2

Simplify .

Step 8.5.2

Perform the row operation to make the entry at a .

Step 8.5.2.1

Perform the row operation to make the entry at a .

Step 8.5.2.2

Simplify .

Step 8.5.3

Multiply each element of by to make the entry at a .

Step 8.5.3.1

Multiply each element of by to make the entry at a .

Step 8.5.3.2

Simplify .

Step 8.5.4

Perform the row operation to make the entry at a .

Step 8.5.4.1

Perform the row operation to make the entry at a .

Step 8.5.4.2

Simplify .

Step 8.5.5

Perform the row operation to make the entry at a .

Step 8.5.5.1

Perform the row operation to make the entry at a .

Step 8.5.5.2

Simplify .

Step 8.5.6

Multiply each element of by to make the entry at a .

Step 8.5.6.1

Multiply each element of by to make the entry at a .

Step 8.5.6.2

Simplify .

Step 8.5.7

Perform the row operation to make the entry at a .

Step 8.5.7.1

Perform the row operation to make the entry at a .

Step 8.5.7.2

Simplify .

Step 8.5.8

Perform the row operation to make the entry at a .

Step 8.5.8.1

Perform the row operation to make the entry at a .

Step 8.5.8.2

Simplify .

Step 8.5.9

Perform the row operation to make the entry at a .

Step 8.5.9.1

Perform the row operation to make the entry at a .

Step 8.5.9.2

Simplify .

Step 8.5.10

Perform the row operation to make the entry at a .

Step 8.5.10.1

Perform the row operation to make the entry at a .

Step 8.5.10.2

Simplify .

Step 8.5.11

Perform the row operation to make the entry at a .

Step 8.5.11.1

Perform the row operation to make the entry at a .

Step 8.5.11.2

Simplify .

Step 8.6

Remove rows that are all zeros.

Step 8.7

Write the matrix as a system of linear equations.

Step 8.8

Since the only solution to is the trivial solution, the vectors are linearly independent.

Linearly Independent

Linearly Independent

Step 9

Since the vectors are linearly independent, they form a basis for the null space of the matrix.

Basis of :

Dimension of :