I don\'t understand question c. Een matrixvergelijking in M2x2(R) Laat A M2x2(R)

Question

I don't understand question c.

Een matrixvergelijking in M2x2(R) Laat A M2x2(R), de vectorruimte van ree..le 2 x 2 matrices. Gegeven is de afbeelding T : M2 times 2(R) rightarrow (R) : X rightarrow AX At. Bewijs dat T een lineaire afbeelding is. Laat B = {M1, M2, M3, M4}, waarbij M1 = (1 0 0 0), M2 = (0 0 1 0), M3 = (0 1 0 0), M4 = (0 0 1 0). Bewijs dat M1, M2, M3, M4 lineair onafhankelijk zijn. Er geldt zelfs dat B een basis is voor M2x2(R), genaamd de standaardbasis. Veronderstel vanaf nu dat A,C M2x2(R) zijn gedefinieerd door A = (2 1 -1 2) en c = (-1 3 3 -9). Geef [A]B en [C]B.

Explanation / Answer

Hello Consider a simpler example. B={(1,0),(0,1)} the standard basis for R2. and an element (2,3) in R2. Since B forms a basis for R2, every element x of R2 can be written as a(1,0)+b(0,2) where a and b are the scalars. In this case, (2,3) = 2(1,0)+3(0,2) Thus the matrix representation of (2,3) is [2,3]T (formed by taking the scalars in the same order as they appear in the linear sum of basis elements) In our case, B = what you wrote, the four matrices. Notice that M1 = element at 1x1 is 1 M2 = element at 2x1 is 1. M3 = element with 1x2 as 1. M4 = element at 2x2 is 1. This means that the matrix representation of A will be [2,-1,1,2]T and that of C will be [-1,3,3,-9]. Notice that [2,-1,-1,2]T when converted back means 1x1 element of matrix is 2, 2x1 element is -1, 1x2 element is -1 and 2x2 element is 2. In general if Basis B ={b1,b2...bn} and x belongs to V. then x = a1b1+a2b2...anbn. Then x is the matrix [a1,a2...an]T. The scalars ai and their order (depending on the order of basis elements bi) depend uniquely on the element x they represent. Hope this was clear and helpful.

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