Answer:
Let B {bı, b2} and C = {c1, c2} be bases for R^2 where bı [-1;8]. b2 [1;-5], c1 = [1;4]. Find the change-of-coordinates matrix from B to C and the change-of-coordinates matrix from C to B.
In the given problem, we have the bases B and C as follows:
B = {b1, b2} where b1 = [-1;8] and b2 = [1;-5]
C = {c1, c2} where c1 = [1;4] and c2 = [0;1]
We need to find the change-of-coordinates matrix from B to C and from C to B.
To find the change-of-coordinates matrix from B to C, we need to express the basis vectors of B in terms of the basis vectors of C. We can do this by solving the following equations:
b1 = x1c1 + y1c2
b2 = x2c1 + y2c2
where x1 , y1, x2, and y2 are the coefficients we want to find.
Substituting the given values, we get:
[-1;8] = x1*[1;4] + y1*[0;1]
[1;-5] = x2*[1;4] + y2*[0;1]
Solving these equations, we get:
x1 = -17/4, y1 = 1/4, x2 = 9/4, and y2 = -1/4
Therefore, the change-of-coordinates matrix from B to C is:
[-17/4 9/4]
[ 1/4 -1/4]
To find the change-of-coordinates matrix from C to B, we need to express the basis vectors of C in terms of the basis vectors of B. We can do this by solving the following equations:
c1 = a1b1 + a2b2
c2 = b1b1 + b2b2
where a1 , a2, b1, and b2 are the coefficients we want to find.
Substituting the given values and solving these equations, we get:
a1 = 4/17, a2 = -1/17, b1 = -1/17, and b2 = -5/17
Therefore, the change-of-coordinates matrix from C to B is:
[ 4/17 -1/17]
[-1/17 -5/17]
I hope this helps!
Explanation: