notes - L A B B Rn stands for all(real n-vectors(columns...

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LABB Rnstands for all (real)n-vectors (columns) andRm×nstands for all (real)m×n-matrices.Usual typographic conventions apply:sR,xRn, andARm×n.We write eitherx·yorhx,yiforxty=x1y1+· · ·+xnyn. The magnitude of a vector iskxk=x·x. Vectorsxandyareorthogonal(notationxy) ifx·y=0.The letterIstands for the identity matrix and the letterJstands for a matrix of all 1s. Tospecify the exact matrix we can writeIn[orJn] to show that we mean ann×nidentity [allones] matrix.0stands for the zero vector and1stands for a vector of all 1s.The transpose of a matrixAis denotedAt.Proposition 1.(AB)t=BtAt.L Vectorsx1, . . . ,xkarelinearly independentiffthe only coefficientsc1, . . . ,cksuch thatc1x1+· · ·+ckxk=0arec1=· · ·=ck=0. Therankof a matrix is the maximum numberof linearly independent columns; notation: rank(A).Proposition 2.Let ARn×m.Thenrank(A)=rank(At)(row rank equals column rank).Furthermore,rank(A)equals the size of a largest invertible (square) submatrix of A.In addition,rank(A)equals the dimension of the subspace{Ax:xRm} ⊆Rn.For square matrices:Theorem 3.Let ARn×n. The following are equivalent:rank(A)=nA is invertible.Ax=0iffx=0.detA,0.The columns of A spanRn.0is not an eigenvalue of A.D

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