transition matrix

Question

B = {2 - 9x, 1 - 4x}

C = {3x - 2, 3x - 3}

for the vector space P2

E = {1, x}

Find the transition matrix from C to B

= ???

Answer 1

Given the 2 × 1 matrices,

⌈2 - 9x⌉ ⌈3x - 2⌉

B = | | and C = | |

⌊1 - 4x⌋ ⌊3x - 3⌋

We can find the matrix T that transforms C to B by setting up the following equation.

T · C = B eq. 1

Because C is 2 × 1 and the right hand side of the equation is the 2 × 1 matrix B, we conclude that the transformation matrix T needs to be 2 × 2. Thus,

⌈ a b ⌉

T = | |

⌊ c d ⌋

We can therefore rewrite eq. 1 as the following.

⌈ a b ⌉ ⌈3x - 2⌉ ⌈2 - 9x⌉

| | · | | = | |

⌊ c d ⌋ ⌊3x - 3⌋ ⌊1 - 4x⌋

By performing the matrix multiplication, we get

3ax - 2a + 3bx - 3b = 2 - 9x

3cx - 2c + 3dx - 3d = 1 - 4x

By simplifying, we get

(3a + 3b)x - (2a + 3b) = -9x + 2

(3c + 3d)x - (2c + 3d) = -4x + 1

By setting the coefficients of x equal in each equation, and by setting the constants equal in each equation, we get

3a + 3b = -9

2a + 3b = -2

and

3c + 3d = -4

2c + 3d = -1

By solving the first system, we get

a = -7, b = 4

And by solving the second system, we get

c = -3, d = 5/3

So the transformation matrix T is

⌈ -7 4 ⌉

T = | |

⌊ -3 5/3 ⌋