Question

How does one calculate a ||f1||??? doing the Gram-Schmidt Algorithm

Answer 1

gram schmidt eh .... sounds orthogonal to me

Answer 2

if you are doing an algorithm, isnt that just a matter of following the instructions?

Answer 3

What do you mean? Like I have the equation and all, I just don't know how my prof calculated ||f1||

Answer 4

From what I can see you just multiple f1 by itself

Answer 5

can you post the problem, to give this a little more context? i know || is a magnitude of a vector but i see no convention of f1

Answer 6

||f1|| would be the square root of : f1 dot f1

Answer 7

Oh lol sorry.
FInd the QR factorization of A = \[\left[\begin{matrix}1 & 3&2 \\ -1&1&0\\0&2&1\\1&1&4\end{matrix}\right]\]

Answer 8

so I followed along an example in my notes and did out to find f1, f2 and f3 with the equations
f1=v1
f2=\[f1\times f1\left( v2 - \frac{ v2 \times f1 }{ f1 \times f1 } \times f1 \right)\]

Answer 9

and f3=\[f1 \times f2 \left( v3-\frac{ v3 \times f1 }{ f1 \times f1 } \times f1 - \frac{ v3 \times f2 }{ f2 \times f2 } \times f2\right)\]

Answer 10

the first vector is generally taken to be the first column vector

\[u_1 =a_1 = [1~-1~0~ 1]\]
\[e_1 =\frac{a_1}{|a_1|}\text{ the unit vector of }a_1\]
\[u_2 =a_2-(a_2\cdot e_1)e_1~~...\]

Answer 11

so I solved for those three... then in my notes I seen that I need to find q1, q2 and q3
by q1 = \[\frac{ f1 }{ ||f1|| }\]
and q2 =\[\frac{ f2 }{ ||f2|| }\]
and q3= \[\frac{ f3 }{ ||f3|| }\]

Answer 12

But I'm not sure what ||f1|| means

Answer 13

Theses are all vectors btw lol

Answer 14

||v|| is notation for the length of a vector \(\vec v\).

any vector, that is divided by its own length, become a unit vector of itself. you can use it as a blank slate to scale as needed

Answer 15

How do you like solve it though? Do you just times the vector by itself?

Answer 16

if youve already determined the vectors for the fs, make them into unit vector

you can acheive the length of a vector by dotting it to itself and taking the sqrt of the results

for example: let v = (3,1,2)

v.v = 3,1,2
3,1,2
------
9+1+4 = 14

|v| = sqrt(14)

Answer 17

square add sqrt is also a good memory devies

Answer 18

Ohhhhh... So for example if f1=(1,-1,0,1) it'd be sqrt(3) ?

Answer 19

yep

Answer 20

Oh okay thank you :)

Answer 21

youre welcome, good luck :)