# 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

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