Linear Algebra (Math 220)
Assignment due Thursday, February 28

1.  Preparation

Expect a quiz.

Relevant Reading:

• Lay § 1.7 and § 4.3

• Hefferon § 2.II

• Matthews §§ 3.3 – 3.4

2.  Exercises

1. Find the inverse of the matrix $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill -4& \hfill 1\\ \hfill -2& \hfill 10& \hfill -1& \hfill 1\\ \hfill 1& \hfill 0& \hfill 1& \hfill 5\\ \hfill 2& \hfill -9& \hfill 1& \hfill 0\end{array}\right)\text{.}$$

2. Let $f$ be the linear map from ${\mathbf{R}}^4$ to ${\mathbf{R}}^4$ that is given by the matrix $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill -4& \hfill 7\\ \hfill -2& \hfill -1& \hfill -1& \hfill -8\\ \hfill -1& \hfill 4& \hfill -14& \hfill 5\\ \hfill 5& \hfill 7& \hfill -11& \hfill 29\end{array}\right)\text{.}$$ In this example it will be found that the reduced row echelon form of the matrix $M$ has only two non-zero rows. We shall come to understand that in this situation both the kernel of $f$ and the image of $f$ are $2$-dimensional.

   1. Obtain a parametric representation for the kernel of $f$.

   2. Find a pair of equations in $4$ variables that characterize the image of $f$.

   3. List a pair of equations in $4$ variables that characterize the kernel of $f$.

   4. Give a parametric representation for the image of $f$.