# Linear Algebra Examples

Step 1
To determine if the columns in the matrix are linearly dependent, determine if the equation has any non-trivial solutions.
Step 2
Write as an augmented matrix for .
Step 3
Find the reduced row echelon form.
Step 3.1
Multiply each element of by to make the entry at a .
Step 3.1.1
Multiply each element of by to make the entry at a .
Step 3.1.2
Simplify .
Step 3.2
Perform the row operation to make the entry at a .
Step 3.2.1
Perform the row operation to make the entry at a .
Step 3.2.2
Simplify .
Step 3.3
Perform the row operation to make the entry at a .
Step 3.3.1
Perform the row operation to make the entry at a .
Step 3.3.2
Simplify .
Step 3.4
Perform the row operation to make the entry at a .
Step 3.4.1
Perform the row operation to make the entry at a .
Step 3.4.2
Simplify .
Step 3.5
Perform the row operation to make the entry at a .
Step 3.5.1
Perform the row operation to make the entry at a .
Step 3.5.2
Simplify .
Step 3.6
Multiply each element of by to make the entry at a .
Step 3.6.1
Multiply each element of by to make the entry at a .
Step 3.6.2
Simplify .
Step 3.7
Perform the row operation to make the entry at a .
Step 3.7.1
Perform the row operation to make the entry at a .
Step 3.7.2
Simplify .
Step 3.8
Perform the row operation to make the entry at a .
Step 3.8.1
Perform the row operation to make the entry at a .
Step 3.8.2
Simplify .
Step 3.9
Perform the row operation to make the entry at a .
Step 3.9.1
Perform the row operation to make the entry at a .
Step 3.9.2
Simplify .
Step 3.10
Multiply each element of by to make the entry at a .
Step 3.10.1
Multiply each element of by to make the entry at a .
Step 3.10.2
Simplify .
Step 3.11
Perform the row operation to make the entry at a .
Step 3.11.1
Perform the row operation to make the entry at a .
Step 3.11.2
Simplify .
Step 3.12
Perform the row operation to make the entry at a .
Step 3.12.1
Perform the row operation to make the entry at a .
Step 3.12.2
Simplify .
Step 3.13
Perform the row operation to make the entry at a .
Step 3.13.1
Perform the row operation to make the entry at a .
Step 3.13.2
Simplify .
Step 3.14
Perform the row operation to make the entry at a .
Step 3.14.1
Perform the row operation to make the entry at a .
Step 3.14.2
Simplify .
Step 3.15
Perform the row operation to make the entry at a .
Step 3.15.1
Perform the row operation to make the entry at a .
Step 3.15.2
Simplify .
Step 4
Remove rows that are all zeros.
Step 5
Write the matrix as a system of linear equations.
Step 6
Since the only solution to is the trivial solution, the vectors are linearly independent.
Linearly Independent