8 determine the matrix which represents t with

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8
Determine the matrix which representsTwith respect to the basis10,11forR2,and100,110,111forR3.(d) Find the least squares solution toAx=111.Solution.(a) We transformAinto echelon form:111001R1R1-R2,R1R1-R3,R1R2,R2R3------------------------→100100Both columns ofAare pivot columns so Nul(A) ={0}and the empty set is a basis forNul(A); and a basis for Col(A) is110,101and110,101-101·110110·110110=110,12-121is an orthogonal basis for Col(A).We transformATinto echelon form:110101R2R2-R1,R2→-R2,R1R1-R2--------------------→10101-1Hence, Nul(AT) = span-111and-111is a basis for Nul(AT). Since the firstand the second column ofAare pivot columns, a basis for Col(AT) is span11,10and11,10-10·1111·1111=11,12-12is an orthogonal basis for Col(AT).(b) LetW= Col(A). We have to find the orthogonal projection of elements of the standardbasis ontoW. The orthogonal projection of the first standard basis vector is:100W=100·110110·110110+100·12-12112-121·12-12112-121=2313139
The orthogonal projection of the second standard basis vector is:010W=010·110110·110110+010·12-12112-121·12-12112-121=1323-13The orthogonal projection of the third standard basis vector is:001W=001·110110·110110+001·12-12112-121·12-12112-121=13-1323Hence, the projection matrix is:2313131323-1313-1323Also, since Col(AT) =R2the projection matrix corresponding to orthogonal projectiononto Col(AT) is the 2×2 identity matrix.

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