Page 113 - 35Linear Algebra
P. 113
6.1 The Consequence of Linearity 113
2
In fact, since every vector in R can be expressed as
x 1 0
= x + y ,
y 0 1
2
we know how L acts on every vector from R by linearity based on just two pieces of
information;
x 1 0 1 0 5 2 5x + 2y
L = L x + y = xL +yL = x +y = .
y 0 1 0 1 3 2 3x + 2y
Thus, the value of L at infinitely many inputs is completely specified by its value at
just two inputs. (We can see now that L acts in exactly the way the matrix
5 2
3 2
2
acts on vectors from R .)
Reading homework: problem 2
This is the reason that linear functions are so nice; they are secretly very
simple functions by virtue of two characteristics:
1. They act on vector spaces.
2. They act additively and homogeneously.
3
A linear transformation with domain R is completely specified by the
way it acts on the three vectors
1 0 0
0 , 1 , 0 .
0 0 1
n
Similarly, a linear transformation with domain R is completely specified
by its action on the n different n-vectors that have exactly one non-zero
component, and its matrix form can be read off this information. However,
not all linear functions have such nice domains.
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