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Lecture02
1
September 06, 2007
Outline:
1) The relationship between systems of linear
equations and
Linear Combinations
of vectors
2) The relationship between systems of equations
and
Ax
=b
3) Basic rules of Matrixvector products
Ax
as linear combinations of vectors
4)
Ax
=b
: the fundamental issues
5) Solving linear systems:
Introduction to Gaussian Elimination
Lecture 02:
Systems of Linear Equations and Ax
=b
Systems of linear Equations
A simple example:
x
 y = 1
2x + y = 5
Solution by "elimination" and "backsubstitution":
The geometry of Linear Systems:
the "row" picture
x
 y = 1
2x + y = 5
x
 y = 1
2x + y = 5
The geometry of Linear Systems:
the "column" picture
Linear systems of equations as Linear Combinations of
vectors
Is equivalent to:
Interpretation: solving linear systems is equivalent to finding linear
combination of vectors that add up to another vector
The geometry of Linear Systems:
the "column" picture
x
 y = 1
2x + y = 5
3x3 problems (3 equations, 3 unknowns):
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 Fall '07
 Spiegelman

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