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

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

egowaffle26ic · Accepted Answer

Step 1  If T(i)=(1, 1) and T(j)=(2, −1) and  e1=i−j=(1, −1) and e2=3i+j=(3, 1), then  T(e1)=T(i−j)  =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 · Accepted Answer

Step 1The matrix of the transformation of basis is:P=[13−11]andP−1=14[1−311]Your transformed matrix is:P−1TP=14[1−311][121−1][13−11]=14[−7−117]Step 2The 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]TIn the new basis we have:v1,2=P−1vi,j=14[−53]and, for the tranformed vector:v1,2′=P−1vi,j=14[21]=P−1Ti.jvi,j=P−1Ti,jPv1,2=T1,2v1,2ao we have:T1,2=P−1Ti,jP=14[−7−117]I hope this can be helpful.

RizerMix · Accepted Answer

Step 1 e1+e2=4i i=e1+e24=P(i|j)T e2−3e1=4j j=e2−3e14=P(i|j)T Now compute the Matrix P−1 and your T in the new Basis is given by P−1TP

Find matrix of linear transformation A linear transformation T:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2} is given by T(i)=i+j T(j)=2i-j

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