Section 3.2 Standard Matrices (AT2)
Learning Outcomes
Translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.
Subsection 3.2.1 Class Activities
Remark 3.2.1.
Recall that a linear map
for any for any
In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.
Activity 3.2.2.
Suppose
Activity 3.2.3.
Suppose
Activity 3.2.4.
Suppose
Activity 3.2.5.
Suppose
The value of
The value of
The value of
Any of the above.
Fact 3.2.6.
Consider any basis
Therefore any linear transformation
Put another way, the images of the basis vectors completely determine the transformation
Definition 3.2.7.
Since a linear transformation
For example, let
Then the standard matrix corresponding to
Activity 3.2.8.
Let
Write the standard matrix
Activity 3.2.9.
Let
(a)
Compute
(b)
Find the standard matrix for
Fact 3.2.10.
Because every linear map
Activity 3.2.11.
Let
(a)
Compute
(b)
Compute
Activity 3.2.12.
Compute the following linear transformations of vectors given their standard matrices.
(a)
(b)
(c)
Subsection 3.2.2 Videos
Subsection 3.2.3 Slideshow
Slideshow of activities available athttps://teambasedinquirylearning.github.io/linear-algebra/2022/AT2.slides.html
.Exercises 3.2.4 Exercises
Exercises available athttps://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/AT2/
.Subsection 3.2.5 Mathematical Writing Explorations
We can represent images in the planeExploration 3.2.13.
For each of the following properties, determine if it is held by the dot product. Either provide a proof it the property holds, or provide a counter-example if it does not.Distributive over addition (e.g., (
Associative?
Commutative?
Exploration 3.2.14.
Given the properties you proved in the last exploration, could the dot product take the place ofExploration 3.2.15.
Is the dot product a linear operator? That is, given vectorsExploration 3.2.16.
Assume
Prove that
Exploration 3.2.17.
A dilation is given by by mapping a vector
to some scalar multiple ofA rotation is given by
A reflection of
over a line can be found by first finding a vector along then
Rotation through an angle
Reflection over a line
passing through the origin.Dilation by some scalar