c-cl-lina.2011t - Linear Algebra MAS4105 6137 Prof JLF King...

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Linear Algebra MAS4105 6137 Class-C Prof. JLF King 1Nov2011 OYOP: For the 2 Essays: Write your grammatical English sentences on every third line, so that I can easily write between the lines. C1: Defn : A collection C := { W 1 , . . . , W 8 } of subspaces is linearly-independent if: . . . Thm: For linear-transformation T : V V , eigenspaces W 1 , . . . , W 8 have ( distinct ) eigenvalues β 1 , . . . , β 8 . Prove that D := { W 1 , . . . , W 8 } is linearly-independent. C2: Matrix M = A B Z D , where A and D are 5 × 5 and 7 × 7, resp., and Z is 7 × 5. Prove that if a GD ( generalized diagonal ) passes through the B block, then it passes through Z . Short answer, OYOP: C3: A system of 3 linear equations in unknowns x 1 , . . . , x 5 reduces to the augmented matrix 5 4 0 0 9 7 0 0 3 0 8 3 0 0 0 2 1 0 , which is almost in RREF. On your own paper, describe the general solution in this form, " x 1 x 2 x 3 x 4 x 5 # = " ? ? ? ? ? # + α " ? ? ? ? ? # + β " ? ? ? ? ? # + γ " ? ? ? ? ? # + . . . where each α, β, γ, δ, . . . is a free variable ( either x 1 or. . . or x 5 ), and each column vector has specific numbers in it. Dim( SolnFlat ) = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C4: 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 Let M ( x ) := 3 x - 2 7 x 4 - 8 10 5 9 x - 2 2 x - 8 8 x x 5 - 2 x 3 + 2 . The high- order term of polynomial Det ( M ( x ) ) is Cx N , where
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