21.14.4. Rank-Nullity Theorem

Try these first

1. 1.

   Define the kernel of a transformation.

   Hint

   Consider what happens to vectors during transformation.

2. 2.

   What is the rank of a linear transformation?

   Hint

   Think about the dimensional space that the transformation maps to.

3. 3.

   What does the Rank-Nullity Theorem establish?

   • sum of rank and nullity is domain dimension
   • null space is always zero
   • rank equals nullity

   Hint

   Remember how rank and nullity relate to each other.

4. 4.

   True or False: The dimension of the kernel is equal to the dimension of the image.

   • True
   • False

   Hint

   Think about how different dimensions affect transformations.

5. 5.

   Given a transformation T from R^5 to R^2, where Ker(T) has dimension 3. Determine the rank and comment on the implications for engineering systems.

   Hint

   Recall how the dimensions interplay via the Rank-Nullity Theorem.

6. 6.

   A civil engineering model uses a transformation T: R^6 -> R^3, with a null space of dimension 4. What is the rank? How does this apply to model predictions?

   Hint

   Apply the Rank-Nullity equation to derive ranks and summaries.