# lec6 - Notes on Linear Algebra A K Lal 2 Chapter 1...

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9/8/2015 1 Module 6: Review of Linear Algebra Vector Spaces and subspaces Linear Independence and Basis Norm and Inner Products Linear Operators Motivation Most State-Space Techniques are Linear Algebraic State equation is a linear operation Modal Decomposition is equivalent to Eigenvalue problem State control and observation involve system of linear equations

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9/8/2015 2 Vector Space A linear vector space X (such as R n or C n ) is a collection of elements called vectors defined over a field that satisfy certain addition and scalar multiplication requirements. Example 1: 3-Dimensional Vectors (R 3 ): x y x+y a x Vector Space Examples Example 2: The set of n-tuples R n (over the field of real numbers). Zero vector, is defined to be a vector with zero elements. Example 3: The set of n-tuples C n (over the field of complex numbers). n n n define n n y x y x , ax ax a y y , x x 1 1 1 1 1 y x x y x