Page 98 - Matrix Analysis & Applied Linear Algebra

92 Chapter 3 Matrix Algebra
Exercises for section 3.3

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2
3.3.1. Each of the following is a function from into . Determine which
are linear functions.

x x x y
(a) f = . (b) f = .
y 1+ y y x
2
x 0 x x
(c) f = . (d) f = 2 .
y xy y y

x x x x + y
(e) f = . (f) f = .
y sin y y x − y

 
x 1
 x 2 
3.3.2. For x =  .  , and for constants ξ i , verify that
.
 . 
x n
f(x)= ξ 1 x 1 + ξ 2 x 2 + ··· + ξ n x n
is a linear function.

3.3.3. Give examples of at least two different physical principles or laws that
can be characterized as being linear phenomena.

2
3.3.4. Determine which of the following three transformations in are linear.
y = x
p

f(p)
f(p)
θ p p

f(p)
Rotate counterclockwise Reflect about Project onto
through an angle θ. the x -axis. the line y = x.