Design a matrix that, when multiplied by a 3D vector ⃗v, swaps the x and y elements and sets z to the average of the three elements of ⃗v.
I have no clue how to do this one any help would be appreciated.
#Linear Algebra
boreal widget
boreal widget
Btw im not asking for the answer. I just need someone get me started and explain how to go about it
storm quail
write the matrix with unknown entries and see what the info get you, you should get 3 equations that should hold for any x, y, z
boreal widget
write the matrix with unknown entries and see what the info get you, you should ...
how do i calculate those equations is there a method i use? we haven't done any examples in class so i have trouble with the start
storm quail
matrix multiplication
boreal widget
with what? jsut random terms?
storm quail
AX = column vector described in the statement
boreal widget
im sorry for the stupid questions but i still dont understand
storm quail
np
A is your matrix, it has to be 3x3 because it sends a column vector of 3 entries to another one
write the entries as unknowns, you want to find them
X = column vecotr [x y z]
it's your generic 'variable vector'
the statement says what is the image AX (as another column vector of 3 entries)
so the matricial equation gives you 3 equations that holds for every X, i.e., for every x, y, z
because you can choose x, y, z, you can get enough equations to determine A
boreal widget
so A is the matrix i'm trying to find. X is supposed to be the column vector where it goes from [x,y,z] -> [y,x, x+y+z/3]. but i still dont understand the later parts do you perhaps have a visual or something i could search up to see how to actually set up my matrix to get those equations
storm quail
like this, multiplied by the column vector X gives the other one you wrote
do the matrix multiplication AX, this gives the left hand sides of the equations
boreal widget
so i do this times x,y,z?
storm quail
storm quail
it's ok, you're learning the conventions and stuff yet it seems
boreal widget
yea its pretty slow in lectures but the hw are on stuff we havent covered
so i multiply the 3 x 3 matrix to the x y z?
a11x+a12y+a13z = y
a21x+a22y+a23z = x
a31x+a32y+a33z = x+y+z/3 like that?
boreal widget
im going to be honest idk if i can do that 😅
storm quail
u mean you don't know why is that valid?
boreal widget
so liek when you say setting values i assign x,y,z random numbers which is always true for the vector we want?
storm quail
it's kinda like you have a function f given by f(x) = x^2 for every x,
so you can set x=1, x=2, x=3 to get f(1) = 1, f(2) = 2^2 = 4, f(3) = 3^2 = 9
AX = that other vector
works for every X, you can set values to it
try x=1, y=z=0 for example and see what you get
boreal widget
Like this?
The right hand terms is the vector we want not the actual result btw incase i wrote it wrong
storm quail
The right hand terms is the vector we want not the actual result btw incase i wr...
it's not wrong
what you want to determine is on the left hand side
boreal widget
how do i do that
storm quail
look at the system you wrote
can you simplify something there? what can you conclude from it?
boreal widget
the first system of equation is 1 = 0 which is false? is that how you mean it
and if i simplify would i apply rref or something
storm quail
the first system of equation is 1 = 0 which is false? is that how you mean it
no, you have a a_{1, 1}*1 there
on the left side
of first equation
storm quail
and if i simplify would i apply rref or something
no i just mean getting rid of the summands that are zero
boreal widget
im sorry im confused
storm quail
your first equation is
a_{1, 1}*1 + a_{1, 2}*0 + a_{1, 3}*0 = 0
can you simplify a_{1, 2}*0? what number can this be?
same for a_{1, 3}*0
then the first
boreal widget
just 0 right?
boreal widget
a_{1, 1}*1 = 1?
boreal widget
so just a11?
boreal widget
and the other two rows can be simplified the same way right
storm quail
ye
storm quail
yes
so that is what you got with the choice
x=1, y=z=0
you can probably guess what other ones can give you the other entries
boreal widget
wdym
storm quail
other choices of values for x, y, z
boreal widget
so now i try like x = 0, y=z=1?
boreal widget
ahh so do i change it to like x and y = 0 and z = 1?
boreal widget
when i plug these in do i also swap for the new values i got for a11,a21,a31
storm quail
you can plug in the values you got for them
but doesn't matter cuz you're setting x=0
boreal widget
how come a23 isnt 1?
and lastly could i x = z = 0, y=1
i apologize for asking so many simple and obvious question i just haven't picked up linear algebra that well
thank you for being so patient and helping me
storm quail
how come a23 isnt 1?
isn't a23 gonna be equal to the value you set for x?
storm quail
and lastly could i x = z = 0, y=1
yep
storm quail
thank you for being so patient and helping me
u're welcome 💙
boreal widget
This is what im writing am i plugging in things incorrectly?
storm quail
ignore the red ones lol
boreal widget
so a23 = 0
storm quail
yeah
boreal widget
but the other values i have written on the left are okay right?
or would a21 be 0 as well
boreal widget
a12 = 1, a22 = 0?
storm quail
yeah
storm quail
2nd and 3rd columns are swapped
you found 3rd column after the first, and then the 2nd column
storm quail
yes
you can check by multiplying it to column[x y z]
ofc it's pretty much what we already done but now they're numbers hehe
by the way
for future reference, if you have a 3x3 matrix A
then A is completely determined by the 3 images
Ae1, Ae2, Ae3
where e1 = column[1 0 0], e2 = column[0 1 0], similar for e3
this is the canonical basis for a 3-dimensional vector space R^3 (or C^3, C complex numbers, or any field really)
Ae1 is the first column of A, Ae2 is the second, Ae3 is the third
this is what we did
when setting x=1, y=z=0 we were calculating Ae1, which is the first column of A
etc
boreal widget
Ohhh ok that makes sense
Ill keep that in mind
Thank you 🙏🏽 for spending hours with me 😭i would have never got this on my own
I appreciate it so much
storm quail
nice that i could help! : )))