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Find qr decomposition [[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]

Solution:
Your problem `->` qr decomposition [[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]


Here `A` = 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`


`q_1'``=a_1` = 
`1`
`1`
`1`
`1`
 = 
`1`
`1`
`1`
`1`


`r_(11)=||q_1'||=sqrt(1^2+1^2+1^2+1^2)=sqrt(4)=2`

`q_1 = 1/(||q_1'||) * q_1'` = `1/2 * `
`1`
`1`
`1`
`1`
 = 
`1/2`
`1/2`
`1/2`
`1/2`


`r_(12)=q_1^T * a_2` = 
[`1/2``1/2``1/2``1/2`]
 `xx` 
`-1`
`4`
`4`
`-1`
`=3`


`q_2'``=a_2-r_(12) * q_1` = 
`-1`
`4`
`4`
`-1`
 -3 
`1/2`
`1/2`
`1/2`
`1/2`
 = 
`-5/2`
`5/2`
`5/2`
`-5/2`


`r_(22)=||q_2'||=sqrt((-2.5)^2+2.5^2+2.5^2+(-2.5)^2)=sqrt(25)=5`

`q_2 = 1/(||q_2'||) * q_2'` = `1/5 * `
`-5/2`
`5/2`
`5/2`
`-5/2`
 = 
`-1/2`
`1/2`
`1/2`
`-1/2`


`r_(13)=q_1^T * a_3` = 
[`1/2``1/2``1/2``1/2`]
 `xx` 
`4`
`-2`
`2`
`0`
`=2`


`r_(23)=q_2^T * a_3` = 
[`-1/2``1/2``1/2``-1/2`]
 `xx` 
`4`
`-2`
`2`
`0`
`=-2`


`q_3'``=a_3-r_(13) * q_1-r_(23) * q_2` = 
`4`
`-2`
`2`
`0`
 -2 
`1/2`
`1/2`
`1/2`
`1/2`
 +2 
`-1/2`
`1/2`
`1/2`
`-1/2`
 = 
`2`
`-2`
`2`
`-2`


`r_(33)=||q_3'||=sqrt(2^2+(-2)^2+2^2+(-2)^2)=sqrt(16)=4`

`q_3 = 1/(||q_3'||) * q_3'` = `1/4 * `
`2`
`-2`
`2`
`-2`
 = 
`1/2`
`-1/2`
`1/2`
`-1/2`


`Q``=[q_1,q_2,q_3]` = 
`1/2``-1/2``1/2`
`1/2``1/2``-1/2`
`1/2``1/2``1/2`
`1/2``-1/2``-1/2`


`R` = 
`r_(11)``r_(12)``r_(13)`
`0``r_(22)``r_(23)`
`0``0``r_(33)`
 = 
`2``3``2`
`0``5``-2`
`0``0``4`




checking `Q xx R = A?`

`Q xx R` = 
`1/2``-1/2``1/2`
`1/2``1/2``-1/2`
`1/2``1/2``1/2`
`1/2``-1/2``-1/2`
 `xx` 
`2``3``2`
`0``5``-2`
`0``0``4`
 = 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`


and `A` = 
`1``-1``4`
`1``4``-2`
`1``4``2`
`1``-1``0`







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