Determinant and area of a parallelogram | Matrix transformations | Linear Algebra | Khan Academy

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Realizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix
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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.
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[@khanacademy]

Thanks. Appreciate your appreciation :)

[@PederBHellandMusic]

Thank you very much! Great video.

[@davidmurphy563]

This seems overly complex. Surely it's just: 1. p = v2.dot(v1) 2. v3 = v1 * p 3. H = v2 - v3 4. Area = v1 * H

[@gracepointks]

Sol, others may have told you this, but I treat your sessions like my DVR. Sometimes I stop you and start over, sometimes I pause you to think about what you have said, sometimes I rewind you a little bit to just confirm what you said and how I thought about it. Also, I sometimes go to other videos that you refer as you go along!

[@D0g63rt]

But how do you prove the cross product then? Do you simply accept it as given or do you wonder where the algorithm for cross product comes from?

[@masterslayer111]

do you teach math? if not you should

[@IrishBog]

So many videos on KZbin just compute this as a given..... it's not intuitive at all that the Det should match the volume but you actually worked it out by hand.... amazing

[@warrenchu6319]

Some teach forming matrix A by using V1 and V2 as row vectors instead of as column vectors. But the determinant of A is the same. Same for 3x3 matrices: det [A] = det [A transposed].

[@akshaybodla163]

For anyone watching this now, if know a bit of calc iii or vector algebra, you can prove ad-bc right from the beginning. We know that the area of the parallelogram is the cross product between the two vectors, in this case v1xv2 which equals x. Since the cross product is more intuitively defined in 3d, then we can write x. We get the terms (0-0)-(0-0)+(ad-cb)=ad-bc. Therefore, the determinant of a 2x2 matrix is ad-bc.

[@BaLiMultiwude]

if you are interested, proceed to develop volume of parallel-piped and other solids area, volume...using that of determinant directly.

[@MercerBay]

Very helpful, thanks!

[@sputternik8]

It seems to me that this can also be proven using the cross product. It was earlier shown that the area of a parallelogram is the length of its vectors' cross product, and the cross product of two R3 vectors is the vector (a2b3-a3b2 , a3b1-a1b3, a1b2-a2b1). If you regard two R2 vectors as R3 vectors with zeros for their third elements, then it simplifies to (0 , 0 , a1b2-a2b1).

[@farhad14]

thanks aloooot i learn frrom ur videos much better than books please keep on. espacially on engineringfield.

[@nachocab834]

This is really helpful. Thanks a lot!

[@bewarethebeef1]

mind = blown

[@charl160]

Excellent video, thanks!

[@beta5770]

Yes. You do know that u derive the cross product of two vectors by constructing a parallelogram

[@Michelle930911]

its is good, but the calculating process is wayyyyy tooo slow! i m assuming ppl here are uni students

[@yustinayasin5539]

wow you're genius, respect!

[@Syeal7]

Did not just interpret your comment sexually. No, I did not.

[@beta5770]

Wow... I never knew how the determinant of a 2x2 matrix is ad - bc. Thanks!

[@Waranle]

Thank you Alot, for taking some time to do all these videos, i really do appreciate :)

[@ingenierox]

Thanks a lot it's very helpful

[@rameshmungamuri36]

i want proof area of the triangle in vector not problem

[@magnus__reeves]

lol English

[@lafyguy]

harry - problem

[@beta5770]

somewhere else as in?

[@jonabirdd]

This is a ridiculously stupid method for computing the area. You could do it in 1/4 the time by subtracting the area outside the parallelogram from the rectangle perpendicular to the coordinate axes that bounds it.

[@debendragurung3033]

This is in the context of linear algebra and will be applicable later. Sometimes Maths is also about proving 1+0=1 but in higher dimension