4.2 Linear Transformations and Isomorphism
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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."