Find a linear mapping G that maps

ropowiec2gkc

ropowiec2gkc

Answered question

2022-04-03

Find a linear mapping G that maps [0,1]×[0,1] to the parallelogram in the xy-plane spanned by the vectors (2,3) and (4,1).

Answer & Explanation

Lana Hamilton

Lana Hamilton

Beginner2022-04-04Added 12 answers

Step 1
Given:
The maps [0,1]×[0,1] to the parallelogram in the XY-plane spanned by the vectors (2,3) and (4,1)
Step 2
Here, basis vectors are (1,0)(0,1) of UV-plane mapping linearly to S and T.
The mapping G(u,v)=(Au+Bv,Cu+Dv) whereA,B, Cand D are constaint.
S=G(1,0)
=(Au+Bv,Cu+Dv)=(A,C)
and
T=G(0,1)
=(Au+Bu,Cu+Du)
=(B,D)
We have given the vectors (2,3) and (4,1) in xy-plane.
G(0,1)=(B,D)N=(2.3)
G(1,0)=(A,C)=(4,1)
That is A=4B=2C=1 and D=3.
So, the liner mapping G that maps [0,1]×[0,1] to the parallelogram in the XY-plane spanned by the vectors
(2,3) and (4,1) is G(u,v)=(4u+2v,u+3v)

mhapo933its

mhapo933its

Beginner2022-04-05Added 9 answers

Step 1
A simpler region in the uv-plane can be linearly mapped to a region described by two vectors r and s.
Step 2
r and s describe a parallelogram region.

Step 3
Recall how the basis vectors i=1,0,j=0,1i of the uv-plane map linearly tor and s. Write the vectors like this to see what is A, B, C, D:
r=G(1,0)=A,B=4,1
s=G(0,1)=C,D=2,3
Then write the mapping:
G(u,v)=(Au+Cv, Bu+Dv)
=(4u+2v, u+3v)
Step 4
It can be confirmed by checking the corners of the uv-rectangle [0,1]×[0,1]:
G(0,0)=(0,0)
G(1,0)=(4,1)
G(0,1)=(2,3)
G(1,1)=(6,4)
Answer
G(u,v)=(4u+2v,u+3v)

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