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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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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
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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Interpretation of
Interpretation of
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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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
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.
 Spring '08
 HOROWITZ

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