Consider the ordered basis B of R2 consisting of the vectors

Consider the ordered basis B of R2 consisting of the vectors [3,7]and [1,1](in that order). Find the vector x in R2 whose coordinates with respect to the basis B are [34]

Solution

The coordinate vectors entries x1,x2 meet the following criterion:

(3, -7)x1 + (-1,1)x2 = (-3, -4)

3x1 - x2 = -3

-7x1 +x2 = -4

------------------( add the two equations)

-4x1 = -7 ----> x1 = 7/4

x2 = 3x1 +3 = 3*7/4 +3 = (21+12)/4 = 33/4

Vector x = ( 7/4 , 33/4)