im(T): Image of a transformation | Matrix transformations | Linear Algebra | Khan Academy

Showing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation.

Missed the previous lesson?

Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

For free. For everyone. Forever. #YouCanLearnAnything

[DevonJones]

WHY ARE THERE NEVER ANY EXAMPLES DONE!!?? The reason why students come to online videos is because they can't understand their teachers because all they demonstrate in class are completely abstract things but this is the same thing!!!

[tolgacandr]

Let me give you an advice you should givve any visible examplese because students get the subjects better with good examples

[vandanaghormade]

You are the best sir... you have the blessing of all the students you have helped...pls keep making this videos...

[jaldabe]

Excelente serie de videos. Muy bien explicados, sin rigorismo matemático pero con los fundamentos claros. Los he disfrutado mucho. Unica sugerenica es la seriación, no resulta fácil encontrar la secuencia correcta.

[Phagocytosis]

No, because although we know that T maps from R^n to R^m, we don't know that it maps to ALL possible vectors in R^m. It could be that whatever vector you take in R^n, if you apply the transformation T to it, there'll be a subset of vectors in R^m that you'll never get to. In that case, the codomain is only part of R^m, not all of it.

[danielgonzalezisaiev]

So since we know that our domain is a subspace (by proving that it is closed under addition and multiplication) and we therefore also know that our co-domain (which is the same thing as the Image of T if I understand it correctly) is a subspace and the task says right from the get go that this transformation T maps from R^n to R^m, can't we just simply - by logical conclusion - say that our co-domain is all of R^m and is therefore also our Image?

Great vid, keep it going!

[lolautoclient]

Does not explain how to actually find the image.

[tinchoneem]

The fact that the 0 vector must be in the space is to avoid the empty set to be a vector space.

[Hateusernamearentu]

so image and range are same thing? really confused.

[danielgonzalezisaiev]

Or to better explain what I'm asking; can we by this logic say that Im(T) is R^m (or perhaps the range of R^m) as soon as we have proven the Image to be a subspace?

[oni]

I can't stop thinking T(V) is a tensor algebra whenever I look at it

[raducumihaicristian]

6:03 . How do we know that T(a) + T(b) = T(a+b). I mean this property would be valid only if T was a Linear Transfsormation. But in this cas T may be just a Transformation (aka a function)

[Thomas-nvsn]

he loves the sound of his own voice more than actually teaching maths

[Aeynx]

@yynotx

Uh yeah, it's linear algebra.

[iRyanP]

It's not redundant to check if it contains zero vector, it may be the empty set.