Question

Problem with linear algebra

Answer 1

columns of P has the length one and they are perpendicular to each other in pairs - explain how these claims can be proven

Answer 2

This can be shown with the dot product and the definition of vector length.

Answer 3

Is it enough to prove that P is a standard basis with rref?

Answer 4

like this

Answer 5

as far as I am aware, the standard basis of the rref is necessarily one, but what can you say about the vector length of the first column entry?

Answer 6

That it is one?

Answer 7

I can show the length of the columns is one with:
\[\left| v \right|=\sqrt{v*v}\]
Right?

Answer 8

exactly, if that's the only thing they require, the perpendicular statement can be validated with the dot product, \[\Large c_1 \cdot c_2=0 \]
where \(c_i\) are the corresponding column vectors.

Answer 9

Great thank you.

Answer 10

you're very welcome.

Answer 11

rref to an identity does not mean that they vectors are perpendicular to one another; it just means that they are independant of each other

Answer 12

to make life easier on the dots, ignore the denominators since each column vector can be scaled to eliminate them

Answer 13

also, if the lengths of the scaled vectors is equal to their denominators, then they have to be unit length