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Problems for Chapter 0: Preliminaries

Basic Skills

1. Choose real numbers \(x_0, y_0\) (arbitrary) and \(a, b, r>0\) and sketch the ellipse and hyperbola: \[ \frac {(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} = r^2 \] \[ \frac {(x - x_0)^2}{a^2} - \frac{(y-y_0)^2}{b^2} = r^2. \] Make notes of how changing \(x_0, y_0\) moves the curves in the plane, and how changing \(a,b\) or \(r\) scales them. What happens if \(r=0\)?

You can check your answers with the following SageMathCell.

2. Let \(\mathbf a\) and \(\mathbf b\) be points in \(\R^n\). Let \(S\) be the set of points in \(\R^n\) whose distance from \(\mathbf a\) is exactly twice their distance from \(\mathbf b\).

   • Write down a formula for \(S\) using mathematical symbols.
   • Sketch \(S\) for the case \(n=2\), \(\mathbf a=(0,0)\), \(\mathbf b = (3,0)\).
   • Prove that \(S\) is actually a sphere of radius \(r\), with centre \({\bf p}\in \R^n\). Find \(r\) and \({\bf p}\). (This is harder, and you should only try it if you think it is interesting.)

3. Given a \(2\times n\) matrix, explain how to determine by inspection whether the two rows are linearly independent.

4. Describe the sets \(\{ \mathbf x\in \R^n : A\mathbf x = {\bf 0} \}\) and \(\{ A\mathbf x : \mathbf x \in \R^n \}\) when \(A\) is a specific \(m\times n\) nonzero matrix, with either \(m=2\) or \(n=2\). Your answers should have the form “This set is a \(\star\)-dimensional subspace of \(\R^*\) when the two rows/columns are linearly dependent/independent.” There are \(4\) cases to consider: \(m=2/n=2\), and dependent/independent.

5. Give a similar description when \(A\) is a nonzero \(1\times n\) matrix or a \(m\times 1\) matrix, that is, a row vector or a column vector.

6. Let \(\mathbf a\) and \(\mathbf u\) be vectors in \(\R^n\), and assume \(|\mathbf u|=1\).

   • Show that there is exactly one number \(t\) such that \(\mathbf a - t \mathbf u\) is orthogonal to \(\mathbf u\). What is \(t\)?
   • Suppose we want to find a number \(t\) such that \(\mathbf a - t \mathbf u\) is orthogonal to \(\bf a\). What condition must \(\mathbf a\) and \(\mathbf u\) satisfy in order for this to be possible?

7. In the figures below, match the level sets on the left with the graphs on the right.

Properties of the dot product and cross product

You should not feel obliged to solve all of these, but we recommend that you try at least a few of them. The ones to try last are questions 11 and 15.

8. Prove Cauchy’s inequality in the special case when both \(\mathbf a\) and \(\mathbf b\) are unit vectors in \(\R^n\), that is, that \(|\mathbf a|^2 = |\mathbf b|^2 = 1\).

   Hint

   Start with \(|\mathbf a-\mathbf b|^2 \ge 0\), and deduce what you can.

9. Prove that for any vectors \(\mathbf a, \mathbf b\in \R^3,\) we have \(\mathbf a\cdot( \mathbf a\times\mathbf b ) = 0.\)

10. Deduce that \(\mathbf b \cdot(\mathbf a \times \mathbf b)=\bf 0\) without redoing the proof.

11. (Scalar triple product.) Given three vectors \(\mathbf a, \mathbf b, {\bf c}\in \R^3\), prove that \[ \mathbf a \cdot (\mathbf b \times {\bf c}) = \det[ \mathbf a, \mathbf b, {\bf c}], \]where \([ \mathbf a, \mathbf b, {\bf c}]\) denotes the matrix whose columns are \(\mathbf a, \mathbf b\), and \({\bf c}\), in that order. Recall that the absolute value of this determinant gives the volume of the parallelepiped in \(\R^3\) with sides given by \(\bf a, b, c\).

12. Let \({\bf r}_1\) and \({\bf r}_2\) be linearly independent row vectors in \(\R^3\), and let \(A\) be the \(2\times 3\) matrix whose rows are \({\bf r}_1\) and \({\bf r}_2\). Let \({\bf v} = {\bf r}_1 \times {\bf r}_2\). Explain why \(\{ \mathbf x\in \R^3 : A \mathbf x = 0 \} = \{ t {\bf v} : t\in \R\}.\) It is fine to give an explanation in words without a detailed mathematical argument.

13. (Reverse triangle inequality). Show that \[ \big| \,|\mathbf a| - |\mathbf b|\, \big| \le |\mathbf a -\mathbf b| \quad \forall \mathbf a, \mathbf b\in \R^n. \]

    Hint

    Start by showing that \(|\mathbf a| - |\mathbf b|\, \le |\mathbf a -\mathbf b|\) for all \(\mathbf a, \mathbf b\).

14. (Cancellation properties). Suppose that \(\mathbf a,\mathbf b, {\bf c}\in \R^3\).

    • Show that if \(\mathbf a\cdot \mathbf b = 0\) and \(\mathbf a\times \mathbf b = {\bf 0}\) then either \(\mathbf a={\bf0}\) or \(\mathbf b = {\bf 0}\).
      Hint

      Use the fact that \(|\mathbf a\times\mathbf b|^2 + (\mathbf a\cdot \mathbf b)^2 = |\mathbf a|^2 |\mathbf b|^2\quad \forall \mathbf a, \mathbf b.\) This was mentioned in Section 0.2.
    • Show that if \({\bf c}\ne {\bf 0}\), \(\mathbf a \cdot{\bf c} = \mathbf b \cdot{\bf c}\) and \(\mathbf a \times{\bf c} = \mathbf b \times{\bf c}\) then \(\mathbf a = \mathbf b\).
      Hint

      Let \(\bf x=\bf a-b\).

15. Prove that \(|\mathbf a\times\mathbf b|^2 = |\mathbf a|^2 |\mathbf b|^2-(\mathbf a\cdot \mathbf b)^2\) using the algebraic definition of the cross product.

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