08-imagekernel

08-imagekernel - IMAGE AND KERNEL Homework: Section 3.1:

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IMAGE AND KERNEL Math 21b, O. Knill Homework: Section 3.1: 10,22,34,44,54,38*,48* IMAGE. If T : R n R m is a linear transformation, then { T ( ~x ) | ~x R n } is called the image of T . If T ( ~x ) = A~x , then the image of T is also called the image of A . We write im( A ) or im( T ). EXAMPLES. 1) If T ( x, y, z ) = ( x, y, 0), then T ( ~x ) = A x y z = 1 0 0 0 1 0 0 0 0 x y z . The image of T is the x - y plane. 2) If T ( x, y ) = (cos( φ ) x - sin( φ ) y, sin( φ ) x + cos( φ ) y ) is a rotation in the plane, then the image of T is the whole plane. 3) If T ( x, y, z ) = x + y + z , then the image of T is R . SPAN. The span of vectors ~v 1 , . . . , ~v k in R n is the set of all combinations c 1 ~v 1 + . . . c k ~v k , where c i are real numbers. PROPERTIES. The image of a linear transformation ~x 7→ A~x is the span of the column vectors of A . The image of a linear transformation contains 0 and is closed under addition and scalar multiplication. KERNEL. If T : R n R m is a linear transformation, then the set { x | T ( x ) = 0 } is called the kernel of T . If T ( ~x ) = A~x , then the kernel of T is also called the kernel of A . We write ker( A ) or ker( T ). EXAMPLES. (The same examples as above) 1) The kernel is the z -axes. Every vector (0 , 0 , z ) is mapped to 0. 2) The kernel consists only of the point (0
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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