Review of vectors and matrices. (The results are stated forvectors and matrices with real entries but also hold for complex entries.)An m × n matrix A ? {aij} is an array of mn numbers:A =a11 a1nam1 amnThis matrix represents the linear mapping (? linear transformation)(1.1)A : Rn ? Rmx ? Ax,where x ? Rn is written as an n × 1 column vector andAx ?am1 amn?x1xn?? ????nj=1 a1jxj.. . nj=1 amjxj???? Rm.The mapping (1.1) clearly satisfies the linearity property:A(ax + by) = aAx + bBy.Matrix addition: If A ? {aij} and B ? {bij} are m × n matrices, then(A + B)ij = aij + bij .Matrix multiplication: If A is m × n and B is n × p, then the matrixproduct AB is the m × p matrix AB whose ij-th element is(1.2) (AB)ij = nk=1aikbkj .Then AB is the matrix of the composition Rp B? Rn A? Rm of th
e twolinear mappings determined by A and B [verify]:(AB)x = A(Bx) ?x ? Rp.1STAT 542 Notes, Winter 2007; MDPTranspose matrix: If A ? {aij} is m × n, its transpose is the n × mmatrix A (sometimes denoted by A) whose ij-th element is aji. That is,the m row vectors (n column vectors) of A are the m column vectors (nrow vectors) of A. Note that [verify](A + B) = A + B (1.3) ;(AB) = BA (1.4) (A : m × n, B : n × p);(A?1) = (A)?1 (1.5) (A : n × n, nonsingular).Rank of a matrix: The row (column) rank of a matrix S : m × n is thedimension of the linear space spanned by its rows (columns). The rank ofA is the dimension r of the largest nonzero minor (= r ×r subdeterminant)of A. Then [verify]row rank(A) ? min(m, n),column rank(A) ? min(m, n),rank(A) ? min(m, n),row rank(A) = m ? dim[row space(A)]?