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Unformatted text preview: 438 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS
(c) L is the least number satisfying Equation (4.8). 5. Find the operator norm L for each linear map L below: (a) L(x, y ) = (y, x).
(b) L(x, y ) = (x + y, x − y ).
(c) L(x, y ) = (x + y 2, x).
(d) L: R2 → R2 is reﬂection across the diagonal x = y .
(e) L: R2 → R3 deﬁned by L(x, y ) = (x, x − y, x + y ). (f) L: R3 → R3 deﬁned by L(x, y, z ) = (x, x − y, x + y ). 4.3 Linear Systems of Equations A system of equations can be viewed as a single equation involving a
mapping. For example, the system of two equations in three unknowns
x +2y +5z = 5
2x + y +7z = 4
can be viewed as the vector equation
L(− ) = −
where L: R3 → R2 is the mapping given by
[L(x, y, z )] = x + 2y + 5z
2x + y + 7z and the right-hand side is the vector with
[− ] =
4 ; we want to solve for the unknown vector with x
[− ] = y .
z We can think of the solution(s) of this system as the “level set” of the
mapping corresponding to the output value − .
y (4.9) ...
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