Thread: Basis problem , finding coordinates

Hi , the problem is written like this:
In a basis (e1,e2,e3) in r^3
F1=(1,0,1) , F2 = (0,1,1) f3 = (1,1,0)
Show That (F1,f2,f3) is also a basis in r^3 and give the relationship Between
the coordinates in both of the basis. Determine the coordinates for e1 + 2*f3



My progress:

Well first i use two fornulas

Y= t^-1 * x (1)

This is the formula to find coordinates of f basis
Where y is the basis f:s coordinates , and where t^-1
is the inverse of the change of basis matrix, and x is. The coordinates which belong
To the basis e

X= t*y. (2) T is the change of basis matrix


So

First of all i put up t

T is. 1 0 1
0 1 1
1 1 0

I have put the rows as columns


I find t through gauss elimination and get

T inverse. Is

1/2. *. 1 -1 1
-1 1 1
1 1 -1


The NeXT step is finding the coordinates

So i begin with formula. (1)

X = e1. And because. E2 and E3 is 0

Therefore the coordinates for x Will be
(1,0,0)

I use x And multiply it with t inverse
And get. (1/2 , -1/2 , 1/2). This is wrong according to the answer
It should be. (1/2, -1/2, 5/2) this in (e)

NeXT i go on using formula (2)

I multiply t with 2f3

And get. (2,2,0)

I add (2,2,0) + (1,0,0) = (3,2,0) this is correct



What am i doing wrong? Is there à better
Way to Solve this problen , pls aid me

Anywa

ok, we have two sets:

and

.

the first task is to show that B' is indeed a basis:

since we know , it suffices to show that B' is linearly independent.

so suppose , that is:



. then:





hence , so thus and thus .

so B' is indeed linearly independent and thus a basis for .

now if T is the matrix that sends (so that ),

then clearly the matrix that turns B-coordinates into B'-coordinates is .

you indeed have found T correctly, and so now we must find T^-1.

we COULD use row-reduction to do this, but an easier way is to express the as linear combinations of the .

now (1,0,0) = (1/2)(1,0,1) + (-1/2)(0,1,1) + (1/2)(1,1,0)

(0,1,0) = (-1/2)(1,0,1) + (1/2)(0,1,1) + (1/2)(1,1,0) and

(0,0,1) = (1/2)(1,0,1) + (1/2)(0,1,1) + (-1/2)(1,1,0), so:



this agrees with what you found.

now B-coordinates are just "the usual cooordinates" that is: .

so the B-coordinates of are: (1,0,0) + 2(1,1,0) = (1,0,0) + (2,2,0) = (3,2,0).

however, the B'-coordinates of will be:

,

so the answer you were given is correct (a sign error perhaps?)

Thank you man, the way you Solve basis and coordinates with, in which book can i read more about it, which book would u recommend, im using à swedish book