kernel and range
DESCRIPTION
teorema de la dimensionTRANSCRIPT
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Kernel and Range
When we talk about the kernel, we are asking for all the vectors in Rn that map to the zero vector in Rm. The range is all the vectors get in Rm. Examples: T: R2 R2, T(x,y) = (x 2y, 3x 6y) Kernel:
To find the kernel, first get the standard matrix associated with the linear transformation. To get the standard matrix of this example, compute T(e1) = T(1,0) and T(e2) = T(0,1).
T(e1) = T(1,0) = (1,3) T(e2) = T(0,1) = (-2,-6)
Now the standard matrix [T] is given by
63
21.
Next, use Gauss-Jordan to find the solution.
=
=
=
=
ty
tx
ty
yxRR
202
00
21
63
21132
So the kernel of this transformation is given by )2,1()2,1()ker( == tT , where t = 1.
Range: To find the range, just take the columns of [T]. Thus the range is given by
=
6
2,
3
1)(Trange
T: R3 R2, T(x,y,z) = (x y + z, 2x + y z) Kernel: Again, to find the kernel we need to find the standard matrix and then put it reduced row-echelon form. T(e1) = T(1,0,0) = (1,2)
T(e2) = T(0,1,0) = (-1,1) T(e3) = T(0,0,1) = (1, -1)
Standard matrix [T]:
112
111
-
Use Gauss-Jordan gives us the following solution:
=
=
=
=
=
=
+
tz
ty
x
tz
zy
x
RRR
RR
0
0
0
110
001
110
111
330
111
112
11121
23
1
122
So the kernel of this transformation is given by: )1,1,0()1,1,0()ker( == tT , where t = 1.
Range: Again to find the range, just take the columns of [T].
=
1
1,
1
1,
2
1)(Trange
Now, we can use the kernel and range to determine their dimension. dim(ker(T)) = nullity(T) = number of free variables
dim(range(T)) = rank(T) = number of leading ones in reduced row-echelon form Example:
512
513
311
To find the dimension of the kernel associated with this matrix, we use Gauss-Jordan.
=
=
=
=
=+
=+
+
tz
ty
tx
tz
zy
zxRRRR
R
RRRRRR
2
0
02
000
110
201
440
110
311
440
110
311
110
440
311
512
513
311
42321
2
32123132
So the dimension of the dim(ker(A)) = 1 and dim(range(T)) = 2. In general there is a formula relating the number of columns of [T] to the dimension of the kernel and the dimension of the range.
dim(ker(T)) + dim(range(T)) = number of columns in [T]