Review for exam 1

Published

September 10, 2025

Our first exam is next Wednesday, September 17! Here’s our first draft of a review sheet.

Problems

  1. State the following definitions:

    1. Pivot position in a matrix. Definition 1.4.1
    2. Linear Combination Definition 2.1.9
    3. The span of a set of vectors Definition 2.3.1
    4. Linear independence of a set of vectors Definition 2.4.5
    5. Homogeneous system of equations Top of section 2.4.3

  2. In this problem, we’re going to consider the types of solutions that might occur and typically do occur for linear systems of various sizes.

    1. Suppose that we have a linear system in 3 equations and 5 unknowns.
      1. Generally, how many solutions do we expect there to be?
      2. Is it possible for there to be a unique solution?
      3. Write down a possible RREF of an augmented matrix for such a system that has no solution.
    2. Suppose that we have a linear system in 5 equations and 3 unknowns.
      1. Generally, how many solutions do we expect there to be?
      2. Is it possible for there to be a unique solution?
      3. Write down a possible RREF of an augmented matrix for such a system that has infinitely many solutions.
    3. Suppose that we have a linear system in 4 equations and 4 unknowns.
      1. Generally, how many solutions do we expect there to be?
      2. Write down a possible RREF of an augmented matrix for such a system that has exactly one solution.

  3. Write down a componentwise proof of the fact that vector addition is associative. That is, if \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are vectors in \(\mathbb{R}^n\), then \[(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}).\]
    I guess you might try the same thing with commutativity.

  1. Consider the vectors that form the columns of the following matrix \(M\): \[ M = \left(\begin{array}{rrrrr} 0 & 1 & 1 & 2 & 2 \\ 2 & -2 & 0 & -2 & 2 \\ -1 & 1 & 0 & 3 & 1 \\ 1 & 3 & 4 & -1 & 1 \end{array}\right) \]

    1. Without doing a single computation, explain why there’s no way for these vectors to be linearly independent.
    2. Now, the RREF of \(M\) is shown below. Based on that, find a linearly independent subset of the columns whose span is the same as the span of all the columns.

\[ \left(\begin{array}{rrrrr} 1 & 0 & 1 & 0 & 2 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) \]

  1. Let \[ \mathbf{x} = \left(\begin{array}{r} 1 \\ 1 \\ 0 \end{array}\right), \: \mathbf{y} = \left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \: \text{and } \mathbf{z} = \left(\begin{array}{r} 0 \\ 0 \\ -1 \end{array}\right). \] Express the vector \[ \mathbf{u} = \left(\begin{array}{r} 2 \\ 3 \\ -4 \end{array}\right) \] as a linear combination of \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\) or explain why there is no such linear combination.

  2. Suppose that \(A\) is a matrix of dimensions \(5\times8\) and \(B\) is a matrix of size \(7\times5\). Then, what are the dimensions of the matrices

    1. \(AB\) and
    2. \(BA\)?

  3. Let \(A\) and \(B\) denote the matrices \[ A = \left(\begin{array}{rr} 3 & 0 \\ 3 & -2 \\ 1 & 0 \end{array}\right) \text{ and } B = \left(\begin{array}{rr} 1 & 0 \\ -3 & -3 \end{array}\right). \]

    1. Compute \(AB\) or explain why that makes no sense.
    2. Compute \(BA\) or explain why that makes no sense.

  4. Write down \(2\times2\) matrices that perform the following actions. In some cases, you might want to express your answer as a product of matrices that perform simpler actions.

    1. Stretches by the factor 2 in the horizontal direction and by the factor 3 in the vertical.
    2. Stretches by the factor 2 in the horizontal direction by the factor 3 in the vertical, and also reflects across the \(x\)-axis.
    3. Projects on the line \(y=x\).
    4. Reflects across a line through the origin that makes an angle of \(17^{\circ}\) with the \(x\) axis.

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