final_review (dragged) 2 - MA 36600 FINAL REVIEW 3 i. A is...

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MA 36600 FINAL REVIEW 3 i. A is invertible i.e., there exists B such that AB = BA = I . ii. A is nonsingular i.e., the x = 0 is the only solution to Ax = 0 . iii. det A ° = 0. Say that we are given an m × n matrix where the entries are functions of time t : A ( t )= ° a ij ( t ) ± We deFne the derivative and (indefnite) integral as those m × n matrices d dt A ( t )= ° d dt a ij ( t ) ± and ² t A ( τ ) = °² t a ij ( τ ) ± . We have the Product Rule i.e., given an m × p matrix A = A ( t ) and a p × n matrix B = B ( t ), d dt ³ A ( t ) B ( t ) ´ = A ( t ) d B dt ( t )+ d A dt ( t ) B ( t ) . We have the Fundamental Theorem o± Calculus i.e., given an m × n matrix A = A ( t ), d dt ² t A ( τ ) = A ( t ) . § 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors. A system of Frst order linear di±erential equations can be expressed in the form d dt x 1 x 2 . . . x m µ ¶· ¸ m -dim’l vector = p 11 ( t ) p 12 ( t ) ··· p 1 n ( t ) p 21 ( t ) p 22 ( t ) ··· p 2 n ( t ) .
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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