Morning,
Bit confused:
I have been given the following:
g(x,y)=(4x,6y)
h(x,y)=(x+2y,y)
k(x,y)=(y,x)
and ive got the following matrix for each of them:
g={{4,0},{0,6}}
h={{1,2},{0,1}}
k={{0,1},{1,0}}
So ive been asked to prove the linear transformation f=k.h.g
has the matrix:
A={{0,6},{4,12}}
but with my matrix work ive done above I get the 6 and the 4 the other way around. Any idea where I went wrong?
Thanks
To add this is the matrix multiplication im doing to get the answer:
{{4,0},{0,6}}*{{0,1},{1,0}}*{{1,2},{0,1}
first h*g
4 12
0 6
then k * h*g
0 6
4 12
in detail
h * g
|1 2| | 4 0 | |4 12|
|0 1| | 0 6 | |0 6 |
then
|0 1| |4 12| |0 6 |
|1 0| |0 6 | |4 12|
Good morning!
Based on the given functions and their corresponding matrices, you have correctly calculated the matrices for each of the functions g, h, and k. However, the order in which you compose these functions matters when calculating the matrix for the composite function f = k ∘ h ∘ g, also denoted as f = k(h(g(x, y))).
To find the matrix representation of the composite function f, you need to multiply the matrices corresponding to the individual functions in the reverse order of their composition. In other words, you need to multiply k, then h, and finally g.
Let's perform the matrix multiplication step-by-step:
1. Multiply h and g to obtain the intermediate result H = h ∘ g:
H = h(g(x, y))
= h(4x, 6y)
= (4x + 2(6y), 6y)
= (4x + 12y, 6y)
The matrix representation for H is:
H = {{1, 2}, {0, 1}} * {{4, 0}, {0, 6}}
= {{4, 8}, {0, 6}}
2. Multiply k with the result H to obtain the final result A = f = k ∘ h ∘ g:
A = k ∘ h ∘ g
= k(H)
= k(4x + 12y, 6y)
= (6y, 4x + 12y)
The matrix representation for A is:
A = {{0, 1}, {1, 0}} * {{4, 8}, {0, 6}}
= {{0, 6}, {4, 8}}
So, the correct matrix representation for the linear transformation f = k ∘ h ∘ g is:
A = {{0, 6}, {4, 8}}
I hope this helps clarify where you went wrong. If you need any further assistance, feel free to ask!