2025-09-26
=====================================================================================

1. If :math:`A` is an invertible matrix, then which of the following statements is **false**?

  a) :math:`det \left( A \right) = 0`

  b) :math:`A x = b` has a unique solution for any vector :math:`b`

  c) :math:`A^T` is invertible

  d) The nullspace of :math:`A` only contains the vector :math:`x = 0`

.. raw:: html

    <details><summary>Answer</summary>a</details>

2. Suppose I have used the Gram-Schmidt process to decompose a square matrix :math:`A` into an orthonormal matrix :math:`Q` and a upper triangular matrix :math:`R` such that :math:`A = Q R`.
Which of the following statements is **true**?

  a) :math:`R^T R = I`

  b) :math:`Q` is a square matrix

  c) :math:`A^{-1} = Q^T R^{-1}`

  d) :math:`q_i \cdot q_j = 1` for any two columns of :math:`Q`

.. raw:: html

    <details><summary>Answer</summary>b</details>

3. Sketch a diagram showing the decomposition of a 2D vector :math:`a` into its projection onto a unit vector :math:`v`, given by :math:`\left( a \cdot v \right) v`, and the projection orthogonal to the unit vector :math:`v`, given by :math:`a - \left( a \cdot v \right) v`.
Assume that :math:`a` is not parallel or orthogonal to :math:`v`.

.. raw:: html

    <details><summary>Answer</summary>See <a href="https://en.wikipedia.org/wiki/Vector_projection">Wikipedia</a> for an example.</details>