MML - Lead up to Exam 2

We will have our second exam Friday, Jan 21. Here are some problem to help whet your appetite. I will probably add a few more problems by our next class period and make a separate, proper review sheet over the weekend.

Note that there is a link at the bottom of this sheet labeled either “Start Discussion” or “Continue Discussion”, depending on the status of those discussions. Following that link will take you to my forum where you can create an account with your UNCA email address. You can ask questions and/or post responses and answers there.

The problems

  1. Define \(U \subset \mathbb R^2\) by \[ U = \{\langle x,y \rangle \in \mathbb R^2: 2x+3y = 0\}. \] Is \(U\) a vector space? If so, then prove that \(U\) is closed under linear combinations. If not then explain clearly why.

  2. Define \(V \subset \mathbb R^2\) by \[ V = \{\langle x,y \rangle \in \mathbb R^2: 2x+3y = 5\}. \] Is \(V\) a vector space? If so, then prove that \(V\) is closed under linear combinations. If not then explain clearly why.

  3. Let \[ A = \begin{bmatrix}1&2\\3&6\end{bmatrix}. \] Give a complete characterization of the range of the linear transformation \(\vec{x} \to A\vec{x}\).

  4. Suppose that the matrix \(A\) and its reduced row echelon form \(R\) are

\[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 3 & 5 & 7 & 9 \end{bmatrix} \quad R = \begin{bmatrix} 1 & 0 & -1 & -2 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

  1. Give a complete description of the column space of \(A\).
  2. Give a complete description of the null space of \(A\).
  3. Give a complete description of the range of \(A\).

  1. Show that the null space of a linear transformation is closed under linear combinations.

  2. Let \[ A = \begin{bmatrix}1&2\\3&5\end{bmatrix}. \]

    1. Row reduce the augmented matrix \([A|I]\).
    2. What is \(A^{-1}\)?

  3. Suppose that \(A\) and \(B\) are invertible matrices of the same size. Show that \((AB)^{-1} = B^{-1}A^{-1}\).

  4. Suppose that the matrix \[ B = \begin{bmatrix} 0 & -2 & 2 & 4 & 10 \\ -2 & -3 & -1 & -1 & -1 \\ 0 & -1 & 1 & 4 & 1 \\ 2 & 3 & 4 & 7 & 4 \\ 2 & 3 & 4 & 5 & 6 \end{bmatrix} \] is row equivalent to \[ D = \begin{bmatrix} 2 & 3 & 4 & 5 & 6 \\ 0 & -1 & 1 & 2 & 3 \\ 0 & 0 & 3 & 4 & 5 \\ 0 & 0 & 0 & -2 & 2 \\ 0 & 0 & 0 & 0 & 4 \end{bmatrix} \] and that the row operations to get from \(B\) to \(D\) are

  • R1<->R5,
  • R1+R4 -> R4,
  • R3+R4 -> R3,
  • R2 - R1 -> R2,
  • 2R3 + R5 -> R5
  • R4 -> -R4
  • R2<->R3.

What is \(\det(B)\)?

  1. Let \(V\) denote the vector space of continuous functions defined on \([0,1]\). Show that the inner product \[ \langle f,g \rangle = \int_0^1 f(x)g(x)\,dx \] preserves linear combinations.