Hi, this is the question from My book, i Will show

My progress aswell

Problem:A vector u has got the coordinates

(1,2) in Two different basis (e1,e2) and (f1,f2).

We know That f1-f2 = e1+e2. Find the

General relationship between the coordinates

In both of the basis

My progress

First of all.

I use Two formulas

(1) y=t(inverse)*x. T = change of basis Matrix

(2) x=t*y

So i try to set up two different change of basis

Matrices one for the basis of e and with f

To be exact i mean e1+2e2 and F1+2f2

So i use these values to create Two different matrices

T(f)= 1 1

-1 2

And t(e) = 1 1

1 2

I have written the basis values from their

Rows to columns and placed them in the

First columns

I need the general relationship

So i assume That i need the inverse

Of the f:s basis and because formula

With the y has got t inverse

Y represents f basis and i assume

That i need to invert the matrix

Of t(f)

Through gaussian elimination

Or determinants i get inverse of t(f)

1/3. * 2 -1

1 1

How can i proceed to get

The general relationship between

The coordinates in the Two basis?

Can someone Solve this for me

The right answer should be

(y1,y2) = (x1, -4*x1 + 3*x2)