28.9. Linear Transformations and Systems of Linear Equations

Try these first

1. 1.

   What is a linear transformation?

   Hint

   Think about how vectors interact under transformations.

2. 2.

   What condition must be met for a solution to exist in a linear system?

   Hint

   Revisit how the image relates to the output of the transformation.

3. 3.

   What does the kernel of a linear transformation represent?

   • Set of all output vectors
   • Set of vectors mapping to zero
   • The transformation function

   Hint

   Consider what happens to vectors under a transformation.

4. 4.

   True or False: A solution exists for all systems of linear equations.

   • True
   • False

   Hint

   Revisit the conditions for the existence of solutions.

5. 5.

   Given a system of equations defined by matrix A, determine if the vector b can be expressed as a linear combination of A's columns. What steps would you take?

   Hint

   Look for free variables indicating multiple solutions or restricted solutions.

6. 6.

   Consider a linear transformation that maps R^2 to R^2. How would you determine the kernel and image geometrically?

   Hint

   Sketch out how vectors transform using the mapping.