# Algebra Examples

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:

First prove the transform preserves this property.

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

Add the two matrices.

Apply the transformation to the vector.

Rearrange .

Rearrange .

Rearrange .

Break the result into two matrices by grouping the variables.

The addition property of the transformation holds true.

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

Multiply by each element in the matrix.

Apply the transformation to the vector.

Simplify each element in the matrix.

Rearrange .

Rearrange .

Rearrange .

Factor each element of the matrix.

Factor element by multiplying .

Factor element by multiplying .

Factor element by multiplying .

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

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

Apply the transformation to the vector.

Rearrange .

Rearrange .

Rearrange .

The zero vector is preserved by the transformation.

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

Linear Transformation