d-cl-lina.2011t

# d-cl-lina.2011t - Linear Algebra MAS4105 6137 e Prof JLF...

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Linear Algebra MAS4105 6137 Class-D Prof. JLF King 16Nov2011 OYOP: For the 2 Essays: Write your grammatical English sentences on every third line, so that I can easily write between the lines. For a square matrix M , the minimum polyno- mial of M [ minpoly ] is the smallest degree monic polynomial Υ M () st. Υ M ( M ) is the zero-matrix. It is always a polynomial-divisor of the charpoly of M . D1: Use , ·i for an inner-product on R -vectorspace V . x1 State the Cauchy-Schwarz Inequality Thm , carefully stating the IFF-condition for equality. x2 Prove the C-S Inequality Thm , using the axioms for inner-product. x3 Triangle Inequality Thm : For all u , w V , we have that k u + w k 6 k u k + k w k . Prove the Triangle-Inequality Thm , using the C-S thm . D2: Suppose S and T are 3 × 3 matrices, with S invertible . Let f () be the charpoly of product ST . Let g () be the charpoly of TS . Carefully prove that f = g . D3: Short answer: Show no work. Please write DNE in a blank if the described object does not exist or if the indicated operation cannot be performed. a A multivariate polynomial, where each monomial has the same degree, is circle monogamous atrocious gregarious homogeneous monic unitary Unitarian utilitarian b Blanks R . So 1 3+2 i = . . . . . . . . . . . + i · . . . . . . . . . . . . And 5 - i 3 + 2 i = . . . . . . . . . . . . . . . + i · . . . . . . . . . . . . . . . . c Let A := [ 1 0 0 0 ] and B := [ 0 1 0 0 ]. Thus
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