Notes

theorom 229

a liunear transofrmation t = fn --> fm

is t(x->) = A(x->) also,

[T] = A = [T(ei->) T(e2->) ... T(en->)])

[T] is called the standard matrix.

Example:
Compute:
[T] fur T: R^3 --> R^2
T([x1,x2,x3]) = [x1,x2]

t([1,0,0]) = [1,0]
t([0,1,0]) = [0,1]
t([0,0,1]) = [0,0]
so:
[T] = [1, 0, 0
0, 1, 0]

Definition 231:

For a rotation ACW by angle q3 the Standard matrix is
[cos O - sin 0
sin O - cos O]

eg
pi/6 = 0
[cos pi/6 - sin pi/6
sin pi/6 - cos pi/6]

= 1/2

Definition 232.
Suppose T: Fn-> Fm is a lin trans.
1) the range of T is. {T(x->): (x->) e Fn} called range(T)
2) The kernel of T is {x-> e Fn : T(x->) =0->}

Prop 233.
Suppose T: Fn -> Fm is a lin translation, Then
1) Range(T) is a subspace of Fm
2) Ker(T) is a subspace of Fn

Definition 232a.:
- The rank of a linear translation is the dimension of Range (T)
- The nullity of a linear translation is the dimensions of Ker (T)

Prop 234:
Suppose T:Fn -> Fm is a linear translation. Then
Range(T) = Col([T]) and
Ker = Null ([T])

Example 235:
Take T: R^3-> R^3,
T[x1, x2,x3]) =

[x1=x2+x3
2x1 +x2+x3
3x1+2x3]

Find [T], Range(T), and Ker (T)