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9/8/20151Module 6: Review of Linear AlgebraVector Spaces and subspacesLinear Independence and BasisNorm and Inner ProductsLinear OperatorsMotivationMost State-Space Techniques are Linear AlgebraicState equation is a linear operationModal Decomposition is equivalent to Eigenvalue problemState control and observation involve system of linear equations
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9/8/20152Vector SpaceA linear vector space X (such as Rnor Cn) is a collection of elements called vectors defined over a field that satisfy certain addition and scalar multiplication requirements.Example 1: 3-Dimensional Vectors (R3):xyx+yaxVector Space ExamplesExample 2: The set of n-tuples Rn(over the field of real numbers).Zero vector, is defined to be a vector with zero elements.Example 3: The set of n-tuples Cn(over the field of complex numbers).nnndefinennyxyx,axaxayy,xx11111yxxyx