Image and Kernel of a Linear Transformation

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Image and Kernel of a Linear Transformation

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Example. Find the kernel of the linear trans-

formation T from R5 to R4 given by the matrix

⎡

⎤

A =

⎢

⎣

1 5 4 3 2

1 6 6 6 6

1 7 8 10 12

1 6 6 7 8

Solution We have to solve the linear system

T(x) = A0 = 0

rref(A) =

⎡

⎢

⎣

⎥

⎦

1 0 −6 0 6

0 1 2 0 −2

0 0 0 1 2

0 0 0 0 0

The kernel of T consists of the solutions of the

system

⎤

⎥

x1 −6x3 +6x5 = 0

x2 +2x3 −2x5 = 0

x4 +2x5 = 0

⎦ .

2.2 Linear Transformation in Geometry Example. 1 Consider a linear ...

Example. Find the kernel of the linear transformation T from R5 to R4 given by the matrix ⎡ ⎤ A = ⎢ ⎣ 1 5 4 3 2 1 6 6 6 6 1 7 8 10 12 1 6 6 7 8 Solution We have to solve the linear system T(x) = A0 = 0 rref(A) = ⎡ ⎢ ⎣ ⎥ ⎦ 1 0 −6 0 6 0 1 2 0 −2 0 0 0 1 2 0 0 0 0 0 The kernel of T consists of the solutions of the system ⎤ ⎥ x1 −6x3 +6x5 = 0 x2 +2x3 −2x5 = 0 x4 +2x5 = 0 ⎦ . 9

The solution are the vectors x = ⎡ ⎢ ⎣ x1 x2 x3 x4 x5 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 6s − 6t −2s + 2t s −2t t where s and t are arbitrary constants . ker(T)= ⎡ ⎢ ⎣ We can write ⎡ ⎢ ⎣ 6s − 6t −2s + 2t s −2t t 6s − 6t −2s + 2t s −2t t ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ = s ⎤ ⎥ ⎦ : s , t arbitrary scalars ⎡ ⎢ ⎣ 6 −2 1 0 0 ⎤ ⎥ ⎦ + t ⎡ ⎢ ⎣ −6 2 0 −2 1 ⎤ ⎥ ⎦

Page 1 and 2: 3.1 Image and Kernal of a Linear Tr
Page 3 and 4: = x1 1 2 + x2 = (x1 + 3x2) See Fi
Page 5 and 6: Fact: Properties of the image (a).
Page 7 and 8: Definition. Kernel The kernel of a
Page 9: Example. Find the kernel of the lin
Page 13 and 14: Fact 3.1.6: Properties of the kerne

Image and Kernel of a Linear Transformation

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