# Linear Algebra Examples

Step 1

The transformation defines a map from to . To prove the transformation is linear, the transformation must preserve scalar multiplication, addition, and the zero vector.

S:

Step 2

First prove the transform preserves this property.

Step 3

Set up two matrices to test the addition property is preserved for .

Step 4

Add the two matrices.

Step 5

Apply the transformation to the vector.

Step 6

Step 6.1

Rearrange .

Step 6.2

Rearrange .

Step 6.3

Rearrange .

Step 7

Break the result into two matrices by grouping the variables.

Step 8

The addition property of the transformation holds true.

Step 9

For a transformation to be linear, it must maintain scalar multiplication.

Step 10

Step 10.1

Multiply by each element in the matrix.

Step 10.2

Apply the transformation to the vector.

Step 10.3

Simplify each element in the matrix.

Step 10.3.1

Rearrange .

Step 10.3.2

Rearrange .

Step 10.3.3

Rearrange .

Step 10.4

Factor each element of the matrix.

Step 10.4.1

Factor element by multiplying .

Step 10.4.2

Factor element by multiplying .

Step 10.4.3

Factor element by multiplying .

Step 11

The second property of linear transformations is preserved in this transformation.

Step 12

For the transformation to be linear, the zero vector must be preserved.

Step 13

Apply the transformation to the vector.

Step 14

Step 14.1

Rearrange .

Step 14.2

Rearrange .

Step 14.3

Rearrange .

Step 15

The zero vector is preserved by the transformation.

Step 16

Since all three properties of linear transformations are not met, this is not a linear transformation.

Linear Transformation