Linear Transformations and Their Properties

U * A

U + A

U - A

U / A

Check if it preserves vector addition

Check if it preserves scalar multiplication

Check if it is reversible

Check if it transforms the zero vector to zero

The result is U + V - A

The result is U + V

The result is U + V + A

The result is U + V + 2A

Because it does not preserve vector addition

Because it does not preserve scalar multiplication

Because it does not transform the zero vector to zero

Because it is not reversible

U + V - A

U + V + 2A

U + V

U + V + A

It is reversible

It preserves vector addition

It transforms the zero vector to a non-zero vector

It preserves scalar multiplication

They become irrelevant

They become easier to calculate

They become more important

They remain unchanged

It should transform to a non-zero vector

It should remain the zero vector

It should transform to any vector

It should transform to a unit vector

Because they are easier to understand

Because they have well-defined properties like eigenvalues and eigenvectors

Because they are more common in real-world applications

Because they are reversible

To explore more non-linear transformations

To compare linear and non-linear transformations

To focus on linear transformations and their properties

To discuss the history of transformations