Answer:
To find the transition matrix from C to B, we need to find the matrix P such that [M]B = P[M]C for any vector M in V.
a. To find P, we need to express the basis vectors of B in terms of the basis vectors of C and arrange the coefficients into a matrix . The basis vectors of B are the matrices [1 0; 0 0] and [0 1; 0 0], and the basis vectors of C are the matrices [2 2; 3 3] and [0 -2; 0 4].
We can express the first basis vector of B in terms of the basis vectors of C as follows:
[1 0; 0 0] = a[2 2; 3 3] + b[0 -2; 0 4]
Solving for a and b, we get:
a = 1/2 b = -1/4
Similarly, we can express the second basis vector of B in terms of the basis vectors of C as:
[0 1; 0 0] = c[2 2; 3 3] + d[0 -2; 0 4]
Solving for c and d, we get:
c = 1/2 d = 1/2
Therefore, the matrix P is:
P = [1/2 1/2; -1/4 1/2]
b. To find the coordinates of M in the ordered basis B, we need to solve the equation [M]B = P[M]C.
We are given that [M]C = [3; 3], so we have:
[M]B = P[M]C [M]B = [1/2 1/2; -1/4 1/2][3; 3] [M]B = [3/2; 9/4]
Therefore , the coordinates of M in the ordered basis B are [3/2; 9/4]. To find M, we need to express these coordinates as a linear combination of the basis vectors of B:
[3/2; 9/4] = e[1 0; 0 0] + f[0 1; 0 0]
Solving for e and f, we get:
e = 3/2 f =
Explanation: