# HWK4
>(Computer numerical methods) 1a, 1c
>1. Find the three-digit decimal floating-point representation of each of the following numbers:
(a) 2312
(b) 0.01277
> &
Gaussian Elimination with Partial Pivoting (Topic from Section 7.3)
Let $\mathrm{A}=[0.1,2.7 ; 1.0,0.5]$ and $\mathrm{b}=[10 ;-6]$
Solve $A x=b$ in 2-digit precision using the following two methods (see what we did in lab 4):
i) Gaussian elimination without pivoting
ii) Gaussian elimination with partial pivoting
Compare the results with the exact solution $\mathrm{x}=[-8 ; 4]$
Show your work
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>1a, 1b
>1. Determine whether the following sets form subspaces of $\mathbb{R}^2$ :
(a) $\left\{\left(x_1, x_2\right)^T \mid x_1+x_2=0\right\}$
(b) $\left\{\left(x_1, x_2\right)^T \mid x_1 x_2=0\right\}$
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>[!note] 4b
>1. Determine the null space of each of the following matrices:
(b) $\left[\begin{array}{rrrr}1 & 2 & -3 & -1 \\ -2 & -4 & 6 & 3\end{array}\right]$
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>1. Determine whether the following are subspaces of $C[-1,1]$ :1
(a) The set of functions $f$ in $C[-1,1]$ such that $f(-1)=$ $f(1)$
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>1. Determine whether the following are spanning sets for $\mathbb{R}^2$ :
(a) $\left\{\binom{2}{1},\binom{3}{2}\right\}$
(b) $\left\{\binom{2}{3},\binom{4}{6}\right\}$
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>1. Given
>
$
\begin{aligned}
& \mathbf{x}_1=\left(\begin{array}{r}
-1 \\
2 \\
3
\end{array}\right), \quad \mathbf{x}_2=\left(\begin{array}{l}
3 \\
4 \\
2
\end{array}\right) \\
& \mathbf{x}=\left(\begin{array}{l}
2 \\
6 \\
6
\end{array}\right), \quad \mathbf{y}=\left(\begin{array}{r}
-9 \\
-2 \\
5
\end{array}\right)
\end{aligned}
$
(a) Is $\mathbf{x} \in \operatorname{Span}\left(\mathbf{x}_1, \mathbf{x}_2\right)$ ?
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