Vector 5 Problem Set 5

  • Due: Friday November 20 by 11:55pm CST.
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The Problem Set covers sections 4.1, 4.2, 4.3 and 4.4.

5.1 A Subspace from Two Linear Transformations

Suppose that \(T: \mathbb{R}^n \to \mathbb{R}^m\) and \(S: \mathbb{R}^n \to \mathbb{R}^m\) are linear transformations. Let \(V \subset \mathbb{R}^n\) be the set \[ V = \{ \mathsf{v} \in \mathbb{R}^n\mid T(\mathsf{v}) = S(\mathsf{v}) \}. \]

Prove that the set \(V\) is a subspace by showing that:

  1. If \(\mathsf{v} \in V\) and \(\mathsf{w} \in V\) then \(\mathsf{v}+\mathsf{w} \in V\)

  2. If \(\mathsf{v} \in V\) and \(c \in \mathbb{R}\) then \(c \mathsf{v} \in V\)

5.2 Getting Into a Subspace

Let \(S \subset \mathbb{R}^n\) be a subspace and let \(\mathsf{v}, \mathsf{w} \in \mathbb{R}^n\). For each of the following statements, either give a specific example or explain why it cannot happen.

  1. If \(\mathsf{v}\) is in \(S\) but \(\mathsf{w}\) is not in \(S\), can \(\mathsf{v} + \mathsf{w}\) be in \(S\)?

  2. If \(\mathsf{v}\) is not in \(S\) and \(\mathsf{w}\) is not in \(S\), can \(\mathsf{v} + \mathsf{w}\) be in \(S\)?

  3. If \(\mathsf{v}\) is not in \(S\) and \(c\) is a nonzero constant, can \(c\mathsf{v}\) be in \(S\)?

5.3 Creating a Basis from Another Basis

Suppose that \(\mathsf{v}_1, \mathsf{v}_2, \mathsf{v}_3\) is a basis of \(\mathbb{R}^3\). Let \[ \mathsf{w}_1 = \mathsf{v}_1, \quad \mathsf{w}_2 = \mathsf{v}_1 + \mathsf{v}_2, \quad \mathsf{w}_3= \mathsf{v}_1 + \mathsf{v}_2 + \mathsf{v}_3. \] Prove that \(\mathsf{w}_1, \mathsf{w}_2, \mathsf{w}_3\) is also a basis for \(\mathbb{R}^3\) as follows:

  1. Show that \(\mathsf{w}_1, \mathsf{w}_2, \mathsf{w}_3\) are linearly independent as follows:
    • We know that whenever \(a_1 \mathsf{v}_1 + a_2 \mathsf{v}_2 + a_3 \mathsf{v}_3 = \mathbf{0}\), this means that \(a_1 = a_2 = a_3 = 0\).
    • Now consider a linear combination \(b_1 \mathsf{w}_1 + b_2 \mathsf{w}_2 + b_3 \mathsf{w}_3 = \mathbf{0}\). Use the previous fact to show that \(b_1 = b_2 = b_3 = 0\).
  2. Show that \(\mathsf{w}_1, \mathsf{w}_2, \mathsf{w}_3\) span \(\mathbb{R}^3\) as follows:
    • We know that for any \(\mathsf{b} \in \mathbb{R}^3\), there exist \(a_1, a_2, a_3 \in \mathbb{R}\) such that \(\mathsf{b} = a_1 \mathsf{v}_1 + a_2 \mathsf{v}_2 + a_3 \mathsf{v}_3\).
    • Use the previous fact to show that there also exist \(c_1, c_2, c_3 \in \mathbb{R}\) such that \(\mathsf{c} = b_1 \mathsf{w}_1 + b_2 \mathsf{w}_2 + b_3 \mathsf{w}_3\).

5.4 A Vector that is in Both Col(A) and Nul}(A)

Give a \(3 \times 3\) matrix \(A\) for which the vector \(\mathsf{v} = \begin{bmatrix}3 \\ -2 \\ 5 \end{bmatrix}\) is in both \(\mathrm{Col}(A)\) and \(\mathrm{Nul}(A)\). Be sure to demonstrate that \(\mathsf{v} \in \mathrm{Col}(A)\) and \(\mathsf{v} \in \mathrm{Nul}(A)\).

5.5 Changing Coordinate Systems

Here are three bases for \(\mathbb{R}^4\):

\[\begin{align} {\cal S} &= \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 0 \\0 \\ 1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}, \\ \\ {\cal B} &= \left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ 1 \\ -1~ \\ -1~ \end{bmatrix}, \quad \begin{bmatrix} 1 \\ -1~ \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 0 \\ 0 \\ 1 \\ -1~ \end{bmatrix} \right\}, \\ \\ {\cal C} &= \left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ 1 \\ -1~ \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ -1~ \\ 0 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ -1~ \\ 0 \\ -1~ \end{bmatrix} \right\}. \end{align}\]

Let \(P_{\cal B}\) be the change-of-coordinates matrix from basis \({\cal B}\) to the standard basis \({\cal S}\), and let \(P_{\cal C}\) be the change-of-coordinates matrix from basis \({\cal C}\) to the standard basis \({\cal S}\).

  1. Find \(P_{\cal B}^{-1}\), which is the change-of-coordinates matrix from the standard basis \({\cal S}\) to basis \({\cal B}\).
  2. Find \(P_{\cal C}^{-1}\), which is the change-of-coordinates matrix from the standard basis \({\cal S}\) to basis \({\cal C}\).
  3. Find the change-of-coordinates matrix from basis \({\cal B}\) to basis \({\cal C}\).
  4. Find the change-of-coordinates matrix from basis \({\cal C}\) to basis \({\cal B}\).
  5. Consider the vector \(\mathsf{v}\) where \[ [\mathsf{v}]_{\cal B} = \begin{bmatrix} ~3.5 \\ -1.0 \\ -0.5 \\ ~1.5 \end{bmatrix}_{\cal B}.\] Find \([ \mathsf{v} ]_{\cal C}.\) Then calculate both \(P_{\cal B} [\mathsf{v}]_{\cal B}\) and \(P_{\cal C} [\mathsf{v}]_{\cal C}\) to confirm that they both produce the same vector \(\mathsf{v} =[\mathsf{v}]_{\cal S}\).

Remark: Both \(\cal{B}\) and \(\cal{C}\) are examples of wavelet bases. Wavelets and similar bases are useful for image processing and image compression.