# Nonsquare matrices as transformations between dimensions | Chapter 8, Essence of linear algebra ## Метаданные - **Канал:** 3Blue1Brown - **YouTube:** https://www.youtube.com/watch?v=v8VSDg_WQlA - **Дата:** 16.08.2016 - **Длительность:** 4:27 - **Просмотры:** 1,940,177 ## Описание A brief footnote on the geometric interpretation of non-square matrices. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. 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If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown ## Содержание ### [0:00](https://www.youtube.com/watch?v=v8VSDg_WQlA) hey everyone I've got another quick footnote for you between chapters today when I've talked about linear transformation so far I've only really talked about Transformations from 2D vectors to other 2D vectors represented with 2x two matrices or from 3D vectors to other 3D vectors represented with 3x3 matrices but several commenters have asked about non-square matrices so I thought I'd take a moment to just show what those mean geometrically by now in the series you actually have most of the background you need to start pondering a question like this on your own but I'll start talking through it just to give a little mental momentum it's perfectly reasonable to talk about Transformations between Dimensions such as one that takes 2D vectors to 3D vectors again what makes one of these linear is that grid lines remain parallel and evenly ### [0:55](https://www.youtube.com/watch?v=v8VSDg_WQlA&t=55s) Output in 3d spaced and that the origin maps to the origin what I have pictured here is the input space on the left which is just 2D space and the output of the transformation shown on the right the reason I'm not showing the inputs move over to the outputs like I usually do is not just animation laziness it's worth emphasizing that 2D Vector inputs are very different animals from these 3D Vector outputs living in a completely separate unconnected space encoding one of these Transformations with a matrix is really just the same thing as what we've done before you look at where each basis Vector lands and write the coordinates of the landing spots as The Columns of a matrix for example what you're looking at here is an output of a transformation that takes I hat to the coordinates 21 -2 and J 0 1 notice this means the Matrix encoding our transformation has three rows and two columns which to you standard ### [1:54](https://www.youtube.com/watch?v=v8VSDg_WQlA&t=114s) 3 x 2 matrix terminology makes it a 3x2 matrix in the language of last video the column space of this Matrix the place where all the vectors land is a 2d plane slicing through the origin of 3D space but the Matrix is still full rank since the number of dimensions in this column space is the same as the number of dimensions of the input space so if you see a 3x2 matrix out in the wild you can know that it has the geometric interpretation of mapping two Dimensions to three dimensions since the two columns indicate that the input space has two basis vectors and the three rows indicate that the landing spots for each of those basis vectors is described with three separate coordinates likewise if you see a 2x3 Matrix with two rows and three columns what do you think that means well the three columns indicate that you're starting in a space that has three basis vectors so we're starting in three dimensions and the two rows indicate that the landing spot for each of those three basis vectors is described with only two coordinates so they must be landing in two dimensions so it's a transformation from 3D space onto the 2D plane a transformation that should feel very uncomfortable if you imagine going through it you could also have a transformation from two Dimensions to one dimension one-dimensional Space is really just the number line so a transformation like this takes in 2D vectors and spits out numbers thinking about grid lines remaining parallel and evenly spaced is a little bit messy due to all of the squish happening here so in this case the visual understanding for what linearity means is that if you have a line of evenly spaced dots it would remain evenly spaced once they're mapped onto the number line one of these Transformations is encoded with a 1x two Matrix each of whose two columns has just a single entry the two columns represent where the basis vectors land and each one of those columns requires just one number the number that basis Vector landed on this is actually a surprisingly meaningful type of transformation with close ties to the dot product and I'll be talking about that next video Until ### [4:05](https://www.youtube.com/watch?v=v8VSDg_WQlA&t=245s) Next video: Dot products and duality Then I encourage you to play around with this idea on your own contemplating the meanings of things like matrix multiplication and linear systems of equations in the context of Transformations between different dimensions have fun --- *Источник: https://ekstraktznaniy.ru/video/16330*