4.2 Linear Transformations and Isomorphism

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Presentation transcript:

4.2 Linear Transformations and Isomorphism http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/More-Vector-Spaces-Isomorphism.topicArticleId-20807,articleId-20795.html

Problem 2 Determine if the transformation is linear.

Solution to problem 2

Example 3

Solution to Example 3

Example 3 Solution d

Isomorphism A Linear Transformation is said to be an Isomorphism if it is invertible. Note: It is common to say that linear space V is isomorphic to the linear space W if there exists an isomorphism (an invertible linear transformation) from V to W. In other words, you can multiply every vector in one space times an invertible matrix to generate the second space.

Problem 2 Revisited Determine if the transformation is an isomorphism:

Solution to Problem 2 Part 2

Determining an Isomorphism If the transformation is linear then:

Example 6 Note: P3 means polynomials of degree 3 or lower

Solution to Example 6

Problem 4 Determine if the transformation is linear. If so find the image and kernel and determine if it is an isomorphism (whether it is invertible). T(M) = det(M) from R2x2 to R

Problem 4 Solution T(M) = det(M) from R2x2 to R

Problem 21 Determine if the transformation is linear. If so find the image and kernel and determine if it is an isomorphism (whether it is invertible).

Problem 21 Solution

Problem 10

Solution to Problem 10

p.170 1-31 odd, 43,45 A mathematician is showing a new proof he came up with to a large group of peers. After he's gone through most of it, one of the mathematicians says,"Wait! That's not true. I have a counter-example!"He replies, "That's okay. I have two proofs."