Linear 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