21.14.2. Matrix Representation

Try these first

1. 1.

   Define the kernel of a linear transformation.

   Hint

   Think about what happens when you apply T to some vectors.

2. 2.

   What is the range of a linear transformation?

   Hint

   Consider all possible images that can be created from a set of inputs.

3. 3.

   What is the kernel of a linear transformation?

   Hint

   Think about what inputs give zero outputs in the transformation.

4. 4.

   True or False: The range of a linear transformation can be larger than the codomain.

   • True
   • False

   Hint

   Consider the relationship between outputs and the broader space.

5. 5.

   Given the matrix A = [[1, 2], [3, 4]], analyze the kernel and range. Determine if the transformation is injective.

   Hint

   Consider both null space and image through operations on matrix A.

6. 6.

   If the transformation T represented by a 4x3 matrix has a kernel dimension of 1, what is the maximum rank of this transformation?

   Hint

   Apply the theorem to deduce the rank from given dimensions.