Answer:
your mom
Step-by-step explanation:
go home look at your mom, that is the answer
Let's make up a linear function: y = mx + b.
Find its inverse:
1) interchange x and y: x = my + b x-b
2) solve this result for y: my = x - b, or y = ----------
m
-1
3) replace y by f (x)
So long as m is not zero, this function is defined for all x.
find the orthogonal projection of v= [19,12,14,-17] onto the subspace W spanned by [ [ -4,-1,-1,3] ,[ 1,-4,4,3] ] proj w (v) = [
12345 [234]
<h2>
Answer:</h2>
Hence, we have:
![proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}]](https://tex.z-dn.net/?f=proj_W%28v%29%3D%5B%5Cdfrac%7B464%7D%7B21%7D%2C%5Cdfrac%7B167%7D%7B21%7D%2C%5Cdfrac%7B71%7D%7B21%7D%2C%5Cdfrac%7B-131%7D%7B7%7D%5D)
<h2>
Step-by-step explanation:</h2>
By the orthogonal decomposition theorem we have:
The orthogonal projection of a vector v onto the subspace W=span{w,w'} is given by:
Here we have:
![v=[19,12,14,-17]\\\\w=[-4,-1,-1,3]\\\\w'=[1,-4,4,3]](https://tex.z-dn.net/?f=v%3D%5B19%2C12%2C14%2C-17%5D%5C%5C%5C%5Cw%3D%5B-4%2C-1%2C-1%2C3%5D%5C%5C%5C%5Cw%27%3D%5B1%2C-4%2C4%2C3%5D)
Now,
![v\cdot w=[19,12,14,-17]\cdot [-4,-1,-1,3]\\\\i.e.\\\\v\cdot w=19\times -4+12\times -1+14\times -1+-17\times 3\\\\i.e.\\\\v\cdot w=-76-12-14-51=-153](https://tex.z-dn.net/?f=v%5Ccdot%20w%3D%5B19%2C12%2C14%2C-17%5D%5Ccdot%20%5B-4%2C-1%2C-1%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%3D19%5Ctimes%20-4%2B12%5Ctimes%20-1%2B14%5Ctimes%20-1%2B-17%5Ctimes%203%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%3D-76-12-14-51%3D-153)
![w\cdot w=[-4,-1,-1,3]\cdot [-4,-1,-1,3]\\\\i.e.\\\\w\cdot w=(-4)^2+(-1)^2+(-1)^2+3^2\\\\i.e.\\\\w\cdot w=16+1+1+9\\\\i.e.\\\\w\cdot w=27](https://tex.z-dn.net/?f=w%5Ccdot%20w%3D%5B-4%2C-1%2C-1%2C3%5D%5Ccdot%20%5B-4%2C-1%2C-1%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%5Ccdot%20w%3D%28-4%29%5E2%2B%28-1%29%5E2%2B%28-1%29%5E2%2B3%5E2%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%5Ccdot%20w%3D16%2B1%2B1%2B9%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%5Ccdot%20w%3D27)
and
![v\cdot w'=[19,12,14,-17]\cdot [1,-4,4,3]\\\\i.e.\\\\v\cdot w'=19\times 1+12\times (-4)+14\times 4+(-17)\times 3\\\\i.e.\\\\v\cdot w'=19-48+56-51\\\\i.e.\\\\v\cdot w'=-24](https://tex.z-dn.net/?f=v%5Ccdot%20w%27%3D%5B19%2C12%2C14%2C-17%5D%5Ccdot%20%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%27%3D19%5Ctimes%201%2B12%5Ctimes%20%28-4%29%2B14%5Ctimes%204%2B%28-17%29%5Ctimes%203%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%27%3D19-48%2B56-51%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cv%5Ccdot%20w%27%3D-24)
![w'\cdot w'=[1,-4,4,3]\cdot [1,-4,4,3]\\\\i.e.\\\\w'\cdot w'=(1)^2+(-4)^2+(4)^2+(3)^2\\\\i.e.\\\\w'\cdot w'=1+16+16+9\\\\i.e.\\\\w'\cdot w'=42](https://tex.z-dn.net/?f=w%27%5Ccdot%20w%27%3D%5B1%2C-4%2C4%2C3%5D%5Ccdot%20%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%27%5Ccdot%20w%27%3D%281%29%5E2%2B%28-4%29%5E2%2B%284%29%5E2%2B%283%29%5E2%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%27%5Ccdot%20w%27%3D1%2B16%2B16%2B9%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cw%27%5Ccdot%20w%27%3D42)
Hence, we have:
![proj_W(v)=(\dfrac{-153}{27})[-4,-1,-1,3]+(\dfrac{-24}{42})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=\dfrac{-17}{3}[-4,-1,-1,3]+(\dfrac{-4}{7})[1,-4,4,3]\\\\i.e.\\\\proj_W(v)=[\dfrac{68}{3},\dfrac{17}{3},\dfrac{17}{3},-17]+[\dfrac{-4}{7},\dfrac{16}{7},\dfrac{-16}{7},\dfrac{-12}{7}]\\\\i.e.\\\\proj_W(v)=[\dfrac{464}{21},\dfrac{167}{21},\dfrac{71}{21},\dfrac{-131}{7}]](https://tex.z-dn.net/?f=proj_W%28v%29%3D%28%5Cdfrac%7B-153%7D%7B27%7D%29%5B-4%2C-1%2C-1%2C3%5D%2B%28%5Cdfrac%7B-24%7D%7B42%7D%29%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cproj_W%28v%29%3D%5Cdfrac%7B-17%7D%7B3%7D%5B-4%2C-1%2C-1%2C3%5D%2B%28%5Cdfrac%7B-4%7D%7B7%7D%29%5B1%2C-4%2C4%2C3%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cproj_W%28v%29%3D%5B%5Cdfrac%7B68%7D%7B3%7D%2C%5Cdfrac%7B17%7D%7B3%7D%2C%5Cdfrac%7B17%7D%7B3%7D%2C-17%5D%2B%5B%5Cdfrac%7B-4%7D%7B7%7D%2C%5Cdfrac%7B16%7D%7B7%7D%2C%5Cdfrac%7B-16%7D%7B7%7D%2C%5Cdfrac%7B-12%7D%7B7%7D%5D%5C%5C%5C%5Ci.e.%5C%5C%5C%5Cproj_W%28v%29%3D%5B%5Cdfrac%7B464%7D%7B21%7D%2C%5Cdfrac%7B167%7D%7B21%7D%2C%5Cdfrac%7B71%7D%7B21%7D%2C%5Cdfrac%7B-131%7D%7B7%7D%5D)
Answer: C: (1.5, 5)
Source: I just took this plato test and got it right
Jose - 96.25=9x+8y
dominic - 85.25=3x+10y
