In this activity, we will do operations on vectors. Furthermore, we find the linear combination of a vector in terms of a given set of vectors.

Suppose that the vectors are linearly independent and is a given vector. This code will find the constants such that if is dependent of $.$ Otherwise, this code will output null.

Given the vectors:

First part: Arithmetic on vectors

Compute the scaled vectors below and put your answers in $,$ respectively.

Compute the different resultants:

Compute the linear combinations:

Second Part: Finding the linear combination of the vector w$0$ in terms of the vectors in terms of the generators w$1,$ w$2,$ and w$3.$

Step $1$: Encode the vectors as column vectors. This is done by puttiing $(\text{'})$ at the end of the vector.

For instance, we write

v$1=[-1,1,-1]\text{'}$

as opposed to just

v$1=[-1,1,-1]$

Step $2$: Augment the given generator vectors into a matrix by using the code

$.$

Step $3$: Augment the matrix A and v$\_0$ and call the matrix Aw$.$

Step $4$: Find the reduce reduced$-$row echelon form of the matrix Av using the code:

rref$($Aw$)$

and put the answer in rrefAw.

Step $5$: Extract the end$-$column of the matrix Av using the code:

Root $=$ rrefAw$($:$,$ end$)$

Step $6$: The obtain the matrix back, use the code

AL $=$ rrefAw$($:$,1$:end$-1)$

Step $8$: The last code below determines whether the constants are found or not.

if AL $==$ eye$($size$($AL$))$

c$1=$ Root$(1)$

c$2=$ Root$(2)$

c$3=$ Root$(3)$

else

display$("$No Roots Found"$)$

c$1=$ NaN;

c$2=$ NaN;

c$3=$ NaN;

end

$\%$FIRST PART

$\%$Encoding and Converting the Vectors as column Vector

v$1=$

v$2=$

v$3=$

$\%$Computation of the scaled vectors

v$4=$

v$5=$

v$6=$

$\%$Computation of resultants

v$7=$

v$8=$

$\%$Computation of linear combinations

v$9=$

v$10=$

v$11=$

$\%$SECOND PART

$\%$Encode the vectors as colum vectors

w$0=$

w$1=$

w$2=$

w$3=$

$\%$Augment the given vectors w$1,$ w$2$ and w$3$ as a single matrix A

$\%$Augment A and w$0$ and express as reduced row echelon form.

$\%$Extract the column matrix and the remaining matrix

if AL $==$ eye$($size$($AL$))$

c$1=$ Root$(1)$

c$2=$ Root$(2)$

c$3=$ Root$(3)$

else

display$("$No Roots Found"$)$

c$1=$ NaN;

c$2=$ NaN;

c$3=$ NaN;

end

$\%$Check of the result

$\%$Compute the linear combination of the vectors w$1,$ w$2,$ and w$3$ using the obtained values for the constants and call the answer w

w $=$

w$0$