lecture_E7_12 - Lecture 12&13 linear equations Linear...

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Linear equations, linear functions Engineering (physical) interpretation of linear equations Problem of inversion Determinants Inversion (square matrices) Independence of vectors Range of a matrix Null space of a matrix Lecture 12 &13: linear equations Reference: book, Chapter 6 Additional material: Lecture_12_***.m (available on bSpace) Steven Boyd, Stanford University, EE263
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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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Linear equations Linear equations: Can be written in matrix form: Or alternatively in compact form
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Example of linear equations Linear circuit (example book, Chapter 6, page 370) 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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What is a linear function? Let be a function. It is said to be linear if This is sometimes called superposition
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Matrix representation of a linear function
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Interpretation of
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Interpretation of
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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.
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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 which results in a polynomial in the coefficients of the matrix.
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