Review for quiz 2

We have our second quiz Wednesday, October 1st. Here’s a first draft of a problem sheet for the quiz. It might be the final draft; I’m not sure.

The problems

1. Write down the definition of each of the following:

   1. Invertible Matrix Definition 3.1.1
   2. Subspace of \(\mathbb R^n\) Definition 3.5.1
   3. Basis
      1. of \(\mathbf R^n\) Definition 3.2.3
      2. of a subspace Definition 3.5.4
   4. Column space of a matrix \(A\) Definition 3.5.6
   5. Null space of a matrix \(A\) Definition 3.5.10

2. For both of the matrices \(A\) and \(B\) below, find its inverse or explain briefly why you know the matrix is not invertible. \[ A = \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{bmatrix} \: \: \: \: \: B = \begin{bmatrix} 1 & a & b \\ 0 & 0 & c \\ 0 & 0 & 1 \\ \end{bmatrix} \]

3. For what values of \(a\) is the following matrix invertible?

\[ A = \begin{bmatrix} a & -1 & 0 \\ 0 & -1 & -1 \\ -1 & 0 & -1 \\ \end{bmatrix} \]

4. The matrix \(A\) together with its reduced row echelon form \(R\) are \[ A = \begin{bmatrix} 7 & -1 & 23 & -4 & -20 \\ 2 & 6 & -6 & 1 & 7 \\ -7 & -1 & -19 & -1 & 3 \\ 0 & 8 & -16 & 1 & 11 \\ -7 & -6 & -9 & 5 & 16 \\ \end{bmatrix} \: \: \text{ and } \: \: R = \begin{bmatrix} 1 & 0 & 3 & 0 & -1 \\ 0 & 1 & -2 & 0 & 1 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}. \]

   1. Is \(A\) invertible?
   2. What is the dimension of the column space of \(A\)?
   3. What is the dimension of the row space of \(A\)?
   4. Find a basis for the column space of \(A\).

Your questions and solutions

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