Image and Kernel of a Linear Transformation
Image and Kernel of a Linear Transformation Image and Kernel of a Linear Transformation
Example. If the function from X to Y is invertible, then image(f) = Y . For each y in Y , there is one (and only one) x in X such that y = f(x), namely, x = f −1 (y). Example. Consider the linear transformation T from R 3 to R 3 that projects a vector orthogonally into the x1 − x2-plane, as illustrate in Figure 4. The image of T is the x1−x2-plane in R 3 . Example. Describe the image of the linear transformation T from R 2 to R 2 given by the matrix Solution T x1 x2 = A x1 x2 A = = 1 3 2 6 1 3 2 6 x1 x2 2
= x1 1 2 + x2 = (x1 + 3x2) See Figure 5. 1 2 3 6 = x1 1 2 + 3x2 Example. Describe the image of the linear transformation T from R 2 to R 3 given by the matrix Solution T x1 x2 = ⎡ ⎢ ⎣ See Figure 6. 1 1 1 2 1 3 A = ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ x1 x2 1 1 1 2 1 3 ⎤ ⎥ ⎦ = x1 ⎡ ⎢ ⎣ 1 1 1 ⎤ 1 2 ⎥ ⎦ + x2 ⎡ ⎢ ⎣ 1 2 3 ⎤ ⎥ ⎦
- Page 1: 3.1 Image and Kernal of a Linear Tr
- Page 5 and 6: Fact: Properties of the image (a).
- Page 7 and 8: Definition. Kernel The kernel of a
- Page 9 and 10: Example. Find the kernel of the lin
- Page 11 and 12: The solution are the vectors x =
- Page 13 and 14: Fact 3.1.6: Properties of the kerne
= x1<br />
<br />
1<br />
2<br />
<br />
+ x2<br />
= (x1 + 3x2)<br />
See Figure 5.<br />
<br />
<br />
1<br />
2<br />
3<br />
6<br />
<br />
<br />
= x1<br />
<br />
1<br />
2<br />
<br />
+ 3x2<br />
Example. Describe the image <strong>of</strong> the linear<br />
transformation T from R 2 to R 3 given by the<br />
matrix<br />
Solution<br />
T<br />
<br />
x1<br />
x2<br />
<br />
=<br />
⎡<br />
⎢<br />
⎣<br />
See Figure 6.<br />
1 1<br />
1 2<br />
1 3<br />
A =<br />
⎤<br />
<br />
⎥<br />
⎦<br />
⎡<br />
⎢<br />
⎣<br />
x1<br />
x2<br />
1 1<br />
1 2<br />
1 3<br />
<br />
⎤<br />
⎥<br />
⎦<br />
= x1<br />
⎡<br />
⎢<br />
⎣<br />
1<br />
1<br />
1<br />
⎤<br />
<br />
1<br />
2<br />
⎥<br />
⎦ + x2<br />
<br />
⎡<br />
⎢<br />
⎣<br />
1<br />
2<br />
3<br />
⎤<br />
⎥<br />
⎦
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