In this assignment, you are required to write a Julia

program to solve the following problems. You will use

Pluto notebooks to implement your solutions

Write a function gaussian$\_$step!$($A::Matrix$\{$Float$64\},k$:: Int$)$ in Julia that

performs one step of Gaussian elimination without pivoting, transforming a matrix

$A^{(k)}$ into $A^{(k+1)}.$ The matrix structure follows the form shown in the image, with

partitions representing rows and columns as they evolve through the elimination

process.

The function takes a matrix $A($of type Matrix$\{$Float$64\})$ and an integer $k,$ repre$-$

senting the current step of the Gaussian elimination process. It updates the matrix

$A$ to reflect the transformation into $A^{(k+1)}.$ The function should not perform any

row exchanges $($no pivoting$).$ Your function should operate in$-$place $($modifying the

matrix directly$).$

Given an upper triangular matrix $Uin{R}^{nn}$ and a vector $bin{R}^n,$ write a Julia

function to solve the system of equations $Ux=b$ using backward substitution. The

function must have the following signature:

function backward$\_$substitution$($U::Matrix$\{$Float$64\},$ b::Vector$\{$Float$64\})$

and it must return a vector.

In question $1,$ you defined the function gaussian$\_$step!, which performs a single

step of Gaussian elimination on a matrix without pivoting. Use this function to

implement full Gaussian Elimination and use the function you defined in question

$2$ to solve a system of linear equations $Ax=b.$ The function signature is

function gaussian$\_$elimination$($A::Matrix$\{$Float$64\},$ b::Vector$\{$Float$64\})$

The function returns the solution $x.$

Given a lower triangular matrix $Lin{R}^{nn}$ and a vector $bin{R}^n,$ write a Julia

function to solve the system of equations $Ly=b$ using forward substitution. The

function must have the following signature:

function forward$\_$substitution$($L::Matrix$\{$Float$64\},$ b::Vector$\{$Float$64\})$

and it must return a vector.