Wrath of Math | One-to-One and Onto Linear Transformations | Linear Algebra @WrathofMath | Uploaded 1 month ago | Updated 4 hours ago
We introduce one to one and onto linear transformations. These are properties you may be familiar with, one to one is also called "injective" and onto is called "surjective". These properties are important, and as we'll see in later lectures, many nice things can be said about linear transformations with either or both of these properties. We'll show several examples of linear transformations that are onto or one-to-one or neither. We'll prove a theorem establishing an equivalence between a linear transformation being one-to-one, being onto, and having the zero space as its kernel (provided the domain and codomain have the same finite dimension). #linearalgebra
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Linear Algebra course: youtube.com/playlist?list=PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG
Linear Algebra exercises: youtube.com/playlist?list=PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc
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0:00 Intro
0:32 Definition of One-to-One and Onto
1:19 Visualize the Definitions
3:14 Zero Kernel is same as One-to-One
6:03 Rotation is One to One and Onto
6:46 Projection onto xy Plane is Neither One to One nor Onto
7:55 Transpose is One to One and Onto
9:13 Zero Transformation is Neither One to One nor Onto
10:01 One to One but not Onto
11:31 An Onto Equivalence
15:14 Conclusion
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Outro music by Ben Watts and is available for channel members.
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We introduce one to one and onto linear transformations. These are properties you may be familiar with, one to one is also called "injective" and onto is called "surjective". These properties are important, and as we'll see in later lectures, many nice things can be said about linear transformations with either or both of these properties. We'll show several examples of linear transformations that are onto or one-to-one or neither. We'll prove a theorem establishing an equivalence between a linear transformation being one-to-one, being onto, and having the zero space as its kernel (provided the domain and codomain have the same finite dimension). #linearalgebra
Join Wrath of Math to get exclusive videos, lecture notes, and more:
youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join
Linear Algebra course: youtube.com/playlist?list=PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG
Linear Algebra exercises: youtube.com/playlist?list=PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc
Get the textbook for this course! amzn.to/43xAWEz
0:00 Intro
0:32 Definition of One-to-One and Onto
1:19 Visualize the Definitions
3:14 Zero Kernel is same as One-to-One
6:03 Rotation is One to One and Onto
6:46 Projection onto xy Plane is Neither One to One nor Onto
7:55 Transpose is One to One and Onto
9:13 Zero Transformation is Neither One to One nor Onto
10:01 One to One but not Onto
11:31 An Onto Equivalence
15:14 Conclusion
★DONATE★
◆ Support Wrath of Math on Patreon: patreon.com/join/wrathofmathlessons
◆ Donate on PayPal: paypal.me/wrathofmath
Outro music by Ben Watts and is available for channel members.
Follow Wrath of Math on...
● Instagram: instagram.com/wrathofmathedu
● TikTok: tiktok.com/@wrathofmathedu
● X: https://x.com/wrathofmathedu
● Facebook: facebook.com/WrathofMath
