Let T:P3(R)→M2×2(R) be a linear transformation, then find the matrix of transformation with respect to standard basis. Also, find the condition number of the matrix of transformation with respect to || ||1.

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Phisecome · Accepted Answer

Step 1Let T:P3(R)→M2×2(R) be a linear transformationTo find:(a) The matrix transformtion with respect to standard basis.(b) The condition number of the matrix of transformation withrespect to ||||1.Step 2(a) Consider a given transformation:T:P3(R)→M2×2(R)be a linear transformationNow, define a transformation:T(a+bx+cx2+dx3)=[abcd]Tye standard basis of vector space P3(R)andM2time2(R) areB={a+bx+cx2+dx3}=[abcd]Further, we haveT(1)=[1000]=1+(0)x+(0)x2+(0)x3T(x)=[0100]=(0)+(1)x+(0)x2+(0)x3T(x2)=[0100]=(0)+(0)x+(1)x2+(0)x3T(x3)=[0100]=(0)+(0)x+(0)x2+(1)x3Now, the matrix transformation w.r.t to the standard basis B isTBB′∼A=[1000010000100001]

Donald Proulx · Accepted Answer

(b) From a part (a): A=[1000010000100001] and A−1=[1000010000100001] Now, the condition number of the matrix A w.r.t norm ||||{1}is||A||1⋅||A−1||1. ||A||1=1+1+1+1=2and||A−1||1=sgrt{1+1+1+1}=2. Further solving, we have Condition number =||A−1||1⋅||A−1||1=(2)⋅(2)=4. ⇒Condition number=4.

Let T:P_3(R)\rightarrow M_{2\times2}(R) be a linear transformation, then f

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