Biostat 216 Homework 4

Due Oct 30 @ 11:59pm

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Q1.

Show the following claims.

  1. If \(\mathcal{S}_1\) and \(\mathcal{S}_2\) are two vector spaces of same order, then their intersection \(\mathcal{S}_1 \cap \mathcal{S}_2\) is a vector space.

  2. If \(\mathcal{S}_1\) and \(\mathcal{S}_2\) are two vector spaces of same order, then their union \(\mathcal{S}_1 \cup \mathcal{S}_2\) is not necessarily a vector space.

  3. The span of a set of \(\mathbf{x}_1,\ldots,\mathbf{x}_k \in \mathbb{R}^n\), defined as the set of all possible linear combinations of \(\mathbf{x}_i\)s \[ \text{span} \{\mathbf{x}_1,\ldots,\mathbf{x}_k\} = \left\{\sum_{i=1}^k \alpha_i \mathbf{x}_i: \alpha_i \in \mathbb{R} \right\}, \] is a vector space in \(\mathbb{R}^n\).

  4. The null space of an matrix \(\mathbf{A} \in \mathbb{R}^{m \times n}\) is a vector space.

Q2.

Let \[ \mathbf{A}_1 = \begin{pmatrix} 1 & 3 & -2 \\ 3 & 9 & -6 \\ 2 & 6 & -4 \end{pmatrix}, \quad \mathbf{A}_2 = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}. \]

  1. Find the matrices \(\mathbf{C}_1\) and \(\mathbf{C}_2\) containing independent columns of \(\mathbf{A}_1\) and \(\mathbf{A}_2\).

  2. Find a rank factorization \(\mathbf{A} = \mathbf{C} \mathbf{R}\) of each of \(\mathbf{A}_1\) and \(\mathbf{A}_2\).

  3. Produce a basis for the column spaces of \(\mathbf{A}_1\) and \(\mathbf{A}_2\). What are the dimensions of those column spaces (the number of independent vectors)? What are the ranks of \(\mathbf{A}_1\) and \(\mathbf{A}_2\)? How many independent rows in \(\mathbf{A}_1\) and \(\mathbf{A}_2\)?

Q3.

  1. Show that an orthocomplement set of a set \(\mathcal{X}\) (not necessarily a subspace) in a vector space \(\mathcal{V} \subseteq \mathbb{R}^m\) \[ \mathcal{X}^\perp = \{ \mathbf{u} \in \mathcal{V}: \langle \mathbf{x}, \mathbf{u} \rangle = 0 \text{ for all } \mathbf{x} \in \mathcal{X}\} \] is a vector space.

  2. Show that, if \(\mathcal{S}\) is a subspace of a vector space \(\mathcal{V} \subseteq \mathbb{R}^m\), then \(\mathcal{S} = (\mathcal{S}^\perp)^\perp\).

Q4.

In this exercise, we show the fact \[ \text{rank}(\mathbf{A} + \mathbf{B}) \le \text{rank}(\mathbf{A}) + \text{rank}(\mathbf{B}) \] for any two matrices \(\mathbf{A}\) and \(\mathbf{B}\) of same size in steps.

  1. Show that the sum of two vector spaces \(\mathcal{S}_1\) and \(\mathcal{S}_2\) of same order \[ \mathcal{S}_1 + \mathcal{S}_2 = \{\mathbf{x}_1 + \mathbf{x}_2: \mathbf{x}_1 \in \mathcal{S}_1, \mathbf{x}_2 \in \mathcal{S}_2\} \] is a vector space.

  2. Show that \(\text{dim}(\mathcal{S}_1 + \mathcal{S}_2) \le \text{dim}(\mathcal{S}_1) + \text{dim}(\mathcal{S}_2)\).

  3. Show that \(\mathcal{C}(\mathbf{A} + \mathbf{B}) \subseteq \mathcal{C}(\mathbf{A}) + \mathcal{C}(\mathbf{B})\).

  4. Conclude that \(\text{rank}(\mathbf{A} + \mathbf{B}) \le \text{rank}(\mathbf{A}) + \text{rank}(\mathbf{B})\).

Q5. Fundamental theorem of ranks

  1. Show that \(\text{rank}(\mathbf{A}) = \text{rank}(\mathbf{A}'\mathbf{A})\) and \(\text{rank}(\mathbf{A}') = \text{rank}(\mathbf{A}\mathbf{A}')\). Hint: we did it in class, using rank-nullity theorem.

  2. Show that \(\text{rank}(\mathbf{A}) = \text{rank}(\mathbf{A}')\) using part 1. Hint: we showed this remarkable result using rank factorization in class. This question asks you to show it using part 1.

Q6.

  1. If \(\mathbf{A}\) and \(\mathbf{B}\) are two matrices with the same number of rows, then \[ \mathcal{C}(\begin{pmatrix} \mathbf{A} \,\,\, \mathbf{B} \end{pmatrix}) = \mathcal{C}(\mathbf{A}) + \mathcal{C}(\mathbf{B}). \]

  2. If \(\mathbf{A}\) and \(\mathbf{C}\) are two matrices with the same number of columns, then \[ \mathcal{R} \left( \begin{pmatrix} \mathbf{A} \\ \mathbf{C} \end{pmatrix} \right) = \mathcal{R}( \mathbf{A} ) + \mathcal{R}( \mathbf{C} ) \] and \[ \mathcal{N} \left( \begin{pmatrix} \mathbf{A} \\ \mathbf{C} \end{pmatrix} \right) = \mathcal{N}( \mathbf{A} ) \cap \mathcal{N}( \mathbf{C} ). \]