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# Chapter 2 - Linear Algebra

## 2.1 Scalars, Vectors, Matrices and Tensors

### Q1 [10 Points, M]
Denote the set of all n-dimensional binary vectors with Cartesian product notation

### Q2 [10 Points, S]
Given the vector
$
\boldsymbol{x}=\left[\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
\end{array}\right]
$, 
and the set 
$
S = \{2, 4\}
$,
obtain the vectors $\boldsymbol{x}_{S}$ and $\boldsymbol{x}_{-S}$

### Q3 [20 Points, S]
Evaluate the following expressions with broadcasting rules,

$$
\left[\begin{array}{lll}
0 & 1 & 2
\end{array}\right]+[5]=
$$

$$
\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]+\left[\begin{array}{lll}
0 & 1 & 1
\end{array}\right]=
$$

$$
\left[\begin{array}{l}
0 \\
1 \\
2
\end{array}\right]+\left[\begin{array}{ll}
0 & 1 & 2
\end{array}\right]=
$$

## 2.2-3 Multiplying Matrices and Vectors and Identity and Inverse Matrices

### Q4 [20 Points, H]
Let $A$ be a $2 \times 2$ matrix, if $A B=B A$ for every $B$ of the size $2 \times 2$, Prove that:
$$
A=\left[\begin{array}{ll}
a & 0 \\
0 & a
\end{array}\right], \
a \in \mathbb{R}
$$


## 2.4 Linear Dependence and Span

### Q5 [10 Points, H]
Prove that if a linear system of equations have two solutions, then it has infinitely many solutions.

### Q6 [5 Points, M]
Given $A x=0$, where $A \in \mathbb{R}^{m \times n}$ is any matrix, and $x \in \mathbb{R}^{n}$ is a vector of unknown variables to be solved, what is the condition such that there is infinitely many solutions?

## 2.5 Norms

### Q7 [15 Points, M]
Prove that **Max Norm** follows these conditions,
$$
\begin{align}
f(\boldsymbol{x})=0 \Rightarrow \boldsymbol{x}=\mathbf{0} \\
f(\boldsymbol{x}+\boldsymbol{y}) \leq f(\boldsymbol{x})+f(\boldsymbol{y}) \\
\forall \alpha \in \mathbb{R}, f(\alpha \boldsymbol{x})=|\alpha| f(\boldsymbol{x})
\end{align}
$$

## 2.6 Special Kinds of Matrices and Vectors

### Q8 [10 Points, M]
Solve the following system of equations,

$$
\frac{1}{2}\left[\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1
\end{array}\right] \left[\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
\end{array}\right] = \left[\begin{array}{c}
1 \\
2 \\
3 \\
4 \\
\end{array}\right]
$$

4a3973de7604531c8a61042c34226df704df2d5c
ipynb
Jupyter Notebook
Ch2_Linear-Algebra/Ch2_Exam1.ipynb
arashash/deep_exercises
2c40802ee367ba9bf1f6fa5dad96cfa1a74e092b
MIT
2020-12-09T10:27:37.000Z
2020-12-09T10:27:37.000Z
Ch2_Linear-Algebra/Ch2_Exam1.ipynb
arashash/deep_exercises
2c40802ee367ba9bf1f6fa5dad96cfa1a74e092b
MIT
Ch2_Linear-Algebra/Ch2_Exam1.ipynb
arashash/deep_exercises
2c40802ee367ba9bf1f6fa5dad96cfa1a74e092b
MIT
Qwen/Qwen-72B
1. YES
2. YES