Image and Kernel of a Linear Transformation
Image and Kernel of a Linear Transformation Image and Kernel of a Linear Transformation
Example. Consider an n × n matrix A. Show that im(A 2 ) is contained in im(A). Hint: To show w is also in im(A), we need to find some vector u st. w = Au. Solution Consider a vector w in im(A 2 ). There exists a vector v st. w = A 2 v = AAv = Au where u = Av. 5
Definition. Kernel The kernel of a linear transformation T (x) = Ax is the set of all zeros of the transformation (i.e., the solutions of the equation Ax = 0. See Figure 9. We denote the kernel of T by ker(T ) or ker(A). For a linear transformation T from R n to R m , • im(T ) is a subset of the codomain R m of T , and • ker(T ) is a subset of the domain R n of T . 6
- 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: Fact: Properties of the image (a).
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- Page 13 and 14: Fact 3.1.6: Properties of the kerne
Definition. <strong>Kernel</strong><br />
The kernel <strong>of</strong> a linear transformation T (x) =<br />
Ax is the set <strong>of</strong> all zeros <strong>of</strong> the transformation<br />
(i.e., the solutions <strong>of</strong> the equation Ax = 0. See<br />
Figure 9.<br />
We denote the kernel <strong>of</strong> T by ker(T ) or ker(A).<br />
For a linear transformation T from R n to R m ,<br />
• im(T ) is a subset <strong>of</strong> the codomain R m <strong>of</strong><br />
T , <strong>and</strong><br />
• ker(T ) is a subset <strong>of</strong> the domain R n <strong>of</strong> T .<br />
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