# Precalculus Examples

Step 1

Step 1.1

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

Step 1.1.1

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

Step 1.1.2

Simplify .

Step 1.2

Perform the row operation to make the entry at a .

Step 1.2.1

Perform the row operation to make the entry at a .

Step 1.2.2

Simplify .

Step 1.3

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

Step 1.3.1

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

Step 1.3.2

Simplify .

Step 1.4

Perform the row operation to make the entry at a .

Step 1.4.1

Perform the row operation to make the entry at a .

Step 1.4.2

Simplify .

Step 1.5

Perform the row operation to make the entry at a .

Step 1.5.1

Perform the row operation to make the entry at a .

Step 1.5.2

Simplify .

Step 2

The pivot positions are the locations with the leading in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: and

Pivot Columns: and

Step 3

The basis for the column space of a matrix is formed by considering corresponding pivot columns in the original matrix. The dimension of is the number of vectors in a basis for .

Basis of :

Dimension of :