21.14. Linear Transformations

Try these first

1. 1.

   Define a linear transformation in your own words.

   Hint

   Focus on what happens with vector addition.

2. 2.

   What is the kernel of a linear transformation?

   Hint

   Think of the output of the transformation.

3. 3.

   What is the definition of a linear transformation?

   Hint

   Remember the vector space properties.

4. 4.

   True or False: The kernel of a linear transformation includes all vectors mapped to the zero vector.

   • True
   • False

   Hint

   Think about how 'zero' is represented in transformations.

5. 5.

   A linear transformation T: R^2 to R^2 is defined by T(x, y) = (3x + y, 2x - y). Calculate the kernel and the range of T.

   Hint

   Complement approaches using systems of equations.

6. 6.

   In a transformation defined by a matrix A = [[1, 0], [0, 2]] mapping R^2, analyze how the dimension changes when restricting the mapping to a line.

   Hint

   Draw diagrams if necessary to visualize transformations.