# Linear Algebra Examples

, ,

Step 1

Assign the set to the name to use throughout the problem.

Step 2

Create a matrix whose rows are the vectors in the spanning set.

Step 3

Step 3.1

Swap with to put a nonzero entry at .

Step 3.2

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

Step 3.2.1

Multiply each element of by 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

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

Step 3.4.1

Multiply each element of by 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

Perform the row operation to make the entry at a .

Step 3.6.1

Perform the row operation to make the entry at a .

Step 3.6.2

Simplify .

Step 4

Convert the nonzero rows to column vectors to form the basis.

Step 5

Since the basis has vectors, the dimension of is .