lecture_E7_Linear_Algebra_2p

lecture_E7_Linear_Algebra_2p - 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 Lecture 8: Linear Algebra Problem of inversion Determinants 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
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2 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
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3 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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4 Interpretation of Interpretation of
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5 Fundamental algebraic operations in MATLAB Some example of fundamental operations on vectors: Fundamental algebraic operations in MATLAB Some example of fundamental operations on vectors: Use dot command directly Or transpose a and then multiply by b.
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6 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 a 2 || || i =1 i || a || 1 = n X i =1 | a i | By default the 2 (Euclidian) norm || a || =m a x i =1 ··· n | a
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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) Consider a square 2 x 2 matrix The determinant of the square matrix is The determinant of a general n x n square matrix is a nonlinear operation The determinant of a general n x n square matrix is a nonlinear operation which results in a polynomial in the coefficients of the matrix. If the determinant of the matrix is non zero, the matrix can be inverted.
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This note was uploaded on 09/07/2011 for the course ENGIN 7 taught by Professor Horowitz during the Spring '08 term at University of California, Berkeley.

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lecture_E7_Linear_Algebra_2p - Lecture 8: Linear Algebra...

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