lecture_E7_Linear_Algebra_6p

lecture_E7_Linear_Algebra_6p - Lecture 8: Linear Algebra...

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1 Linear equations, linear functions Engineering (physical) interpretation of linear equations Fundamental linear algebra operations in MATLAB Problem of inversion Determinants Lecture 8: Linear Algebra Inversion (square matrices) Independence of vectors Range of a matrix Null space of a matrix Linear equations Example of linear equations (Chap. 6, p. 365) Which can be written in matrix form Or more symbolically as: Where Linear equations Linear equations: Can be written in matrix form: Or alternatively in compact form Example of linear equations Linear circuit Equations for voltages and intensities are linear In the following equation, if you know the voltages, the intensities can be obtained by solving the following linear system where the unknown are the currents i’s What is a linear function? Let be a function. It is said to be linear if This is sometimes called superposition Matrix representation of a linear function
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2 Interpretation of Interpretation of Fundamental algebraic operations in MATLAB Some example of fundamental operations on vectors: Fundamental algebraic operations in MATLAB Use dot command directly Some example of fundamental operations on vectors: Use dot command directly Or transpose a and then multiply by b. Fundamental algebraic operations in MATLAB Orthogonality of two vectors Scalar product of two vectors is zero implies they are orthogonal Fundamental algebraic operations in MATLAB Norm of two vectors || a || 2 = v u u t n X i =1 a 2 i By default the 2 (Euclidian) norm || a || 1 = n X i =1 | a i | || a || =m a x i =1 ··· n | a
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3 Problem of inversion If all quantities involved were numbers, solving the following problem i.e. finding as a function of and Could formally be written as Of course, the definition of the “inverse”, or “one over” the matrix has to be defined properly, and the conditions in which it is legal for the inverse to exist need to be defined as well. Determinants (square matrices)
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lecture_E7_Linear_Algebra_6p - Lecture 8: Linear Algebra...

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