Matrix condition for one-to-one trans | Matrix transformations | Linear Algebra | Khan Academy

Showing that the rank of the of an mxn transformation matrix has to be an for the transformation to be one-to-one (injective)

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

[kasra.b]

Very nice video but you could have mentioned that to be "one to one", the matrix needs to be linearly independent at the beginning of the video so that those who know what that means don't have to sit through 17mins of lead up.

[beegdigit]

Khan always has that eye opening simplicity

[aidawall]

your handwriting is beautiful i love it!!!
so aesthetically pleasing :)

oh, also the video helped a lot. THANKS!

[ryanlafferty]

onto=surjective, one to one = injective, both = bijective

[blakekl]

What program do you use to draw for these videos?

[foolio]

is this about injective and surjective transformations

[ReusableDuck]

Why can't my professor teach like

[Baradur]

Whoa, I'm surprised the CC functionality of youtube is working so well.

[nicholasgigliotti]

My book says that a matrix can be one to one if it has a unique solution or no solution however, it has says T is one to one if and only if Ax=b has trivial solutions. How can something that only has trivial solutions have no solution. I guess what I am saying is can somebody provide me with a matrix that is one-to-one with no solution?
 

[nenadilic]

3:40 Mistake. b' is in Rm and x particular must be in Rn. They clearly are not the same. Another, smaller, mistake at 15:00 - the arrow in the implication has to point the opposite way: if N(A) is NOT only the zero-vector then T is not 1-to-1, and that is what he had proven. The way the implication was shown simply stems from replacing xn with the zero-vector which leaves only one x (xp) mapping to Ax.

[NGBigfield]

These videos are so good!
and: "let's do this in white!".... "That's not white!!"