08-imagekernel

# 08-imagekernel - IMAGE AND KERNEL Homework Section 3.1...

This preview shows page 1. Sign up to view the full content.

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 , 0 , 0). 3) The kernel consists of all vector ( x, y, z ) for which x + y + z = 0. The kernel is a plane. PROPERTIES. The kernel of a linear transformation contains 0 and is closed under addition and scalar multiplication.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern