---
title: Biostat 216 Homework 4
subtitle: '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
$$
\mathcal{N} \left( \begin{pmatrix} \mathbf{A} \\ \mathbf{C} \end{pmatrix} \right) = \mathcal{N}( \mathbf{A} ) \cap \mathcal{N}( \mathbf{C} ).
$$