v$1=[$

$2$

$1$

$1$

$],$ v$2=[$

$1$

$1$

$0$

$],$ v$3=[$

$0$

$1$

$1$

$]$

what is wrong with my code, i keep getting: Orthonormalized Vectors $($Q$)$:$[[8\mathrm{.}16496581$e$-01-3\mathrm{.}14018492$e$-165\mathrm{.}77350269$e$-01][4\mathrm{.}08248290$e$-017\mathrm{.}07106781$e$-01-5\mathrm{.}77350269$e$-01][-4\mathrm{.}08248290$e$-017\mathrm{.}07106781$e$-015\mathrm{.}77350269$e$-01]].$ When i should get vector$1=[0\mathrm{.}81649658,0\mathrm{.}4824829,-0\mathrm{.}40824829],$ vector$2=[0,0\mathrm{.}70710678,0\mathrm{.}707106678]$ and vector $3=[0\mathrm{.}57735027,-0\mathrm{.}57735027,0\mathrm{.}57735027].$ I included an image of the the expected numbers should be given these certain vectors.

this is my code: import numpy as np

# Define the input vectors

v$1=$ np$.$array$([2,1,-1])$

v$2=$ np$.$array$([1,1,0])$

v$3=$ np$.$array$([0,-1,1])$

# Combine the vectors into a matrix where each column is a vector

V $=$ np$.$column$\_$stack$(($v$1,$ v$2,$ v$3))$

def gram$\_$schmidt$($V$)$:

$$ # Input: V is a set of vectors stored as columns in a matrix

$$ V $=$ V$.$astype$($np$.$float$64)$# Ensure float precision

$$ n$,$ k $=$ V$.$shape

$$ Q $=$ np$.$zeros$\_$like$($V$)$# Matrix to store the orthonormalized vectors

$$ for j in range$($k$)$:

$$ v$\_$j $=$ V$[$:$,$ j$].$copy$()$# Work with the j$-$th vector

$$ for i in range$($j$)$:

$$ q$\_$i $=$ Q$[$:$,$ i$]$

$$ s $=$ np$.$dot$($v$\_$j$,$ q$\_$i$)$# s $$v$\_$j$,$ q$\_$i$$

$$ v$\_$j $=$ v$\_$j $-$ s $*$ q$\_$i $$# v$\_$j $$ v$\_$j $-$ s $*$ q$\_$i

$$ norm$\_$v$\_$j $=$ np$.$linalg.norm$($v$\_$j$)$

$$ if norm$\_$v$\_$j $>1$e$-10$: $$# Normalize to avoid division by zero

$$ Q$[$:$,$ j$]=$ v$\_$j $/$ norm$\_$v$\_$j $$# v$\_$j $$ v$\_$j $/||$v$\_$j$||$

$$ return Q

def verify$\_$orthonormality$($Q$)$:

$$ # Verify magnitudes $($should be close to $1)$

$$ for idx, q in enumerate$($Q$.$T$)$:

$$ magnitude $=$ np$.$linalg.norm$($q$)$

$$ print$($f$"$Magnitude of vector $\{$idx$+1\}=\{$magnitude:$\mathrm{.}5$f$\}")$

$$ print$("-"*50)$

$$ # Verify dot products between different vectors $($should be close to $0)$

$$ k $=$ Q$.$shape$[1]$

$$ for i in range$($k$)$:

$$ for j in range$($i $+1,$ k$)$:

$$ dot$\_$product $=$ np$.$dot$($Q$[$:$,$ i$],$ Q$[$:$,$ j$])$

$$ print$($f$"$Dot product of vectors $\{$i$+1\}$ and $\{$j$+1\}$: $\{$dot$\_$product:$\mathrm{.}5$f$\}")$

# Perform Gram$-$Schmidt orthonormalization

Q $=$ gram$\_$schmidt$($V$)$

# Display the orthonormalized vectors

print$("$Orthonormalized Vectors $($Q$)$:$")$

print$($Q$)$

# Verify orthonormality

print$("$

Running verification function:"$)$

verify$\_$orthonormality$($Q$)$