The Matrix of a Linear Transformation 1.9

0.0(0)
studied byStudied by 0 people
learnLearn
examPractice Test
spaced repetitionSpaced Repetition
heart puzzleMatch
flashcardsFlashcards
Card Sorting

1/15

encourage image

There's no tags or description

Looks like no tags are added yet.

Study Analytics
Name
Mastery
Learn
Test
Matching
Spaced

No study sessions yet.

16 Terms

1
New cards

Any linear transformation can be written as..

A matrix equation

<p>A matrix equation</p>
2
New cards

Theorem 10: Let T: R^n R^m be a linear transformation. Then there exist…

A unique matrix A such that T(x) = Ax for all x in R^n

3
New cards

What is the difference between linear transformation and matrix transformation?

The term linear transformation focuses on a property of a mapping, while matrix transformation describes how such mapping is implemented

4
New cards
<p>Reflection through the x1 axis</p>

Reflection through the x1 axis

<p></p>
5
New cards
<p>Reflection through the x2 axis</p>

Reflection through the x2 axis

knowt flashcard image
6
New cards
<p>Reflection through the line x2 = x1</p>

Reflection through the line x2 = x1

knowt flashcard image
7
New cards
<p>Reflection through the line x2 = -x1</p>

Reflection through the line x2 = -x1

knowt flashcard image
8
New cards
<p>Reflection through the origin </p>

Reflection through the origin

knowt flashcard image
9
New cards
<p>Horizontal contraction and expansion </p>

Horizontal contraction and expansion

knowt flashcard image
10
New cards
<p>Vertical contraction and expansion </p>

Vertical contraction and expansion

knowt flashcard image
11
New cards
<p>Horizontal shear</p>

Horizontal shear

knowt flashcard image
12
New cards
<p> Vertical shear</p>

Vertical shear

knowt flashcard image
13
New cards

Onto Transformation

A transformation T: R^n R^m is said to be onto R^n if each b in R^m is the image at least one x in R^n

<p>A transformation T: R^n <span data-name="arrow_right" data-type="emoji">➡</span>R^m is said to be <strong>onto</strong> R^n if each <strong>b </strong>in R^m is the image <em>at least one </em><strong>x </strong>in R^n</p>
14
New cards

How do we know if the transformation is onto?

Theorem 4

  • Ax=b has a solution for all b in R^m

  • Each b is a linear combination of the columns of A

  • The columns of A span R^m

  • Matrix A has a pivot position in every row

15
New cards

One to one transformation

A transformation T: R^n R^m is one to one if each vector b in R^m is the image of at most 1 vector x in R^n

<p>A transformation T: R^n <span data-name="arrow_right" data-type="emoji">➡</span>R^m is one to one if each vector <strong>b </strong>in R^m is the image of <em>at most </em>1 vector <strong>x </strong>in R^n</p>
16
New cards

How do we know if the transformation is one to one?

The solution has to be a unique solution

  • No free variables

  • Pivot positions in every column

  • The equation T(x) = 0 has only the trivial solution

  • The columns of A are linearly independent