You are required to:

Develop a MATLAB function that:

Accepts a set of vectors $($either in $R^2$ or $R^3).$

Applies the Gram$-$Schmidt process to produce an orthonormal basis.

Handles edge cases such as linearly dependent vectors $($e$.$g$.,$ two vectors pointing in the same direction$).$

Document the theory behind your code, clearly explaining each step of the Gram$-$Schmidt process as implemented in your function. Use inline comments and a separate write$-$up if necessary.

Test your function with multiple input sets, including:

A set of standard basis vectors.

A set of random vectors.

A set of linearly dependent vectors.

Analyze the results and compare them to your theoretical expectations.

Guidelines:

Function Signature: Your function should be defined as follows:

matlab

Copiar c$$digo

function $[$U$]=$ OrthoBasis$($V$)$

$\text{'}$ V $\text{'}$ is an $nm$ matrix where $n$ is the number of vectors and $m$ is the dimension $($either $2$ or $3).$

is an $nm$ matrix containing the orthonormal basis vectors.

Implementation Steps:

Start by normalizing the first vector.

Iteratively apply the Gram$-$Schmidt process to orthogonalize and normalize each subsequent vector.

Ensure your function can handle both $2$D and $3$D cases.

Documentation:

Include detailed comments in your code explaining each operation.