Find matrix of linear transformation A linear transformation T:R2→R2 is given by T(i)=i+j T(j)=2i−j

Emerson Barnes

Emerson Barnes

Answered

2022-01-20

Find matrix of linear transformation
A linear transformation
T:R2R2
is given by
T(i)=i+j
T(j)=2ij

Answer & Explanation

egowaffle26ic

egowaffle26ic

Expert

2022-01-21Added 7 answers

Step 1
If T(i)=(1, 1) and T(j)=(2, 1) and
e1=ij=(1, 1) and e2=3i+j=(3, 1), then
T(e1)=T(ij)
=T(i)T(j)
=(1, 2)
=a1e1+b1e2 (say)
and
T(e2)=T(3i+j)
=3(1, 1)+(2, 1)
=(5, 2)
=a2e1+b2e2 (say)
Thus the matrix of T w.r.t. the new basis {e1, e2} is
(a1a2b1b2)
and you need to find the values of a1, a2, b1 and b2. The above systems of equations reduces to
a1+2b1=1
a1+b1=2
and
a2+3b2=5
a2+b2=2
Solve these equations to obtain
a1=74
b1=14
a2=14
and
b2=74
suzzzlesv7

suzzzlesv7

Expert

2022-01-22Added 7 answers

Step 1
The matrix of the transformation of basis is:
P=[1311]
and
P1=14[1311]
Your transformed matrix is:
P1TP=14[1311][1211][1311]
=14[7117]
Step 2
The figure illustrate how operate the given transformation in the two basis.
Indices {i, j} refer to the canonical basis, indices {1, 2} to the new basis e1, e2.
In the figure we have the vector v=vi,j=[1,2]T (in canonical basis), that is transformed in v=vi,j=[5,1]T
In the new basis we have:
v1,2=P1vi,j=14[53]
and, for the tranformed vector:
v1,2=P1vi,j=14[21]
=P1Ti.jvi,j
=P1Ti,jPv1,2
=T1,2v1,2
ao we have:
T1,2=P1Ti,jP=14[7117]
I hope this can be helpful.
RizerMix

RizerMix

Expert

2022-01-27Added 437 answers

Step 1 e1+e2=4i i=e1+e24=P(i|j)T e23e1=4j j=e23e14=P(i|j)T Now compute the Matrix P1 and your T in the new Basis is given by P1TP

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