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Now that we've covered reshaping and transposing you might be thinking when would we ever want to use

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a transpose.

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It just flips the axis is around.

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When is that useful.

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Well that's what we're going to cover in this video.

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And one of the main use cases for transposing a matrix is the dot product.

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It might be thinking now dot product.

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What the hell is a dot product.

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Well let's do what we do.

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Dot product name pi.

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Let's have a research because what we're practicing.

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Here we go.

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Name pi dot dot product of two arrays.

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Oh that's not very helpful.

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Sometimes you get that then docs it'll give you a very brief explanation of what happens specifically

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if both A and B a one day arise.

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It is the inner product of vectors.

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Okay.

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There's a few text instructions here of what dot product actually is but let's focus on the code.

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I'll go back to our notebook we'll write some code first and then we'll have a little look into what

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the DOT PRODUCT actually does.

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The thing you need to remember the dot product it's just another way of finding patterns between two

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different sets of numbers.

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So let's start off with the random seed will create two more matrices that we can use for an example.

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I want this one to be Rand and let's make it of size 5 and 3 and we want map to beautiful.

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We'll just make it the same thing.

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Random dot Rand int 10.

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Size equals 5 3.

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And now we've seen Rand in before.

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Do you know what these two lines of code will do.

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So let's have a look.

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As a reminder we can do a shift tab return random in is from low to high.

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Beautiful.

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Let's see these in action.

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MATT one wonderful Matt to even more wonderful.

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And then we also want to say we already know the shape but let's just do it anyway.

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So we've got the shapes here.

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Matt to shape wonderful.

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So in a previous video we've seen a few different ways of performing arithmetic on matrices like this.

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One of the ways that we did was multiplication.

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So we go.

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Matt one Times Matt.

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Two.

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Let's bring these down so we can see them because I want to stay nice and zoomed in to keep the code

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big.

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There we go.

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Matt one Matt two wonderful this array down here is just going to be the combination of these two multiplied

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element wise five times six is 30 0 times 7 0 but this is I said a little tricky word before element

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y's multiplication.

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There's a few different names to this but it's mostly called element y's hard Ahmad product.

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That's that's how you spell out I believe.

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Let's have a look.

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Hard hard.

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Product Name pi.

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That's what we're after How to Get element wise.

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Yeah.

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MP dot multiply which is the same as using this little star symbol.

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So that's element y's multiplication.

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Another form of multiplication or product between two matrices is the dot product.

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Let's have a look at that dot product.

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If we go back actually to our documentation the formula for doing a dot product is MP dot.

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You can use num piled up we've imported num pies and P so we're gonna do MP dot form mat 1.

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Map 2 which is just our two matrices that we made.

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So let's run this oh that didn't work.

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Damn it.

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Our value era shapes 5 3 5 3.

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Not aligned.

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3 dim 1 does not equal 5.

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Okay we go back up to see the shapes of our matrices.

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They're both of shape.

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5 3 5 3.

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This is saying shapes 5 3 and 5 3 not aligned well the difference between DOT PRODUCT AND element wise

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so we've seen element wise before it's relatively simple we've got a times a goes into this top left

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square a Times F no sorry b times.

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I just said simple and here I am messing it up the Times F but if we kept going through that we'd eventually

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end up with this matrix.

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But the dot product is a little bit different now we've got say a three by three matrix here a three

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by two matrix here and we're calling dot on this one with this one.

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Now what is going on here the resulting matrix is gonna be three by two and we have a times J.

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Okay.

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Following along with that then we have b times L plus.

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Okay then we have C times n all right.

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So it's got the top row here multiplied by the first column and adding up the total.

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Now let's look at a little bit more of a colorful demonstration of the dot product so here we've got

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the letters matrix if we go back here that the same one is here now we've got some color added.

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This is a top row of the matrix and the left multiplied by the column of the matrix on the right equals

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this top left square.

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And then if we kept going through we would eventually get this resulting matrix here.

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But there's a couple of rules with a dot product that numbers on the inside must match.

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So for a dot product to happen between two matrices they're dimensions.

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If you put them side by side.

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So this three matches this three.

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So this matrix here has three rows and so does this one.

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And now the resulting matrix is the size of the outside numbers.

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So the dimensions here are three by two of this resulting matrix because the outside numbers of these

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two matrices are 3 and 2.

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Okay.

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So we've got that for a dot product to happen.

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The inside numbers must match.

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Now let's have a look with some numbers that have actually been filled in.

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We've got 5 0 3 4 6 8 and we're calling dot between this matrix and this one so we've got 5 it's going

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to come over here be multiplied by 4 that equals 20.

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Yep we've got that 5 4.

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This is a Times J.

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Now we've got 0 times 6 0 times 6.

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This is B times l yep we've got that.

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We're adding that up now we've got three times eight.

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This is C times n Orion and that's 24.

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And if we add it these three numbers together.

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So we're going to go eight times J plus B times l policy times n.

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Yeah beautiful.

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We get 44 and that's going to fill the top left square of our resulting array.

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Now if we went through these two matrices with numbers full and followed the same steps that we've got

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in this letters matrix we would end up with something like this but this is a static demonstration so

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let's check out this little beautiful resource here which is an interactive demo.

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So this is matrix multiplication dot x y z.

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This is called the Waterfall technique.

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What this demonstration is going to do and you can change these numbers here to be something similar

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to maybe what we've got here or just to view the process of what's happening so if we click multiply

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it's going to take the matrix that was on the right and put it on top.

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This where the waterfall process comes in.

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Now it's going to flow down so this row two six one used to be here as a column.

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And if we add these together multiply them we get 15 in the top left.

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What it's done is it's just replicated this step here.

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We've got the top left.

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Now if we follow it through okay we get the top right and we get the middle left and we get the middle

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right and the bottom left and then we finally finish up with the bottom right number.

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So have a play around with this if you check it out you can keep going through it takes a little bit

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of practice to understand the dot product.

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Let's go back to our notebook and now that we know a little bit more about the dot product.

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We know the rules the numbers on the inside much matched and the new size of the resulting matrix is

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three by two because it takes the outside numbers.

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How would we make this work with our matrix 1 and Matrix 2 if their internal shapes are 5 and 3 and

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5 and 3

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or maybe we could use a transpose let's try that out transpose map 1 because what does it transpose

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do.

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It flips the accesses around so let's go map one dot t.

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What does that look like.

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Mm hmm.

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Okay.

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Might look at that and see maybe it'll work but let's really confirm it map one dot t dot shape.

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3 5.

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Is that what we need.

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Yeah.

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We need to flip it around.

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So if we have met two dot shape

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beautiful so now the inside shapes.

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If we were to transpose mat 1 matching so let's do this actually before we did MP dot map one not two.

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Okay maybe we transpose Matt to that one dot shape Matt to t to stay consistent while we go above.

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Let's make Matt three Eagles MP dot map one and then we want Matt to dot t.

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So this should work if our shapes are aligned Matt.

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Three beautiful.

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And now if we have Matt three what is the shape of Matt three yes.

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Excellent.

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So now we can see where the transpose might come into play if we wanted to do a dot product on two matrices

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where in our first try the shapes were incorrect.

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We might want to flip the accesses of course you could do this with the reshape as well but this is

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just a handy example where transpose comes into play for performing something like a dot product.

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Now we can see the resulting shape of Matrix 3 is the same as the outside numbers.

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Wonderful.

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Now when I first came across dot product it took me a little while to understand how to actually go

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through it.

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A great exercise was to write out a matrix by hand and multiply it and figure something out like this

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or use the demo at like Matrix Multiplication done X Y Z.

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The main takeaway from the dot product is that it's just another tool in our arsenal to find patterns

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between two different arrays of numbers.

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As we've seen before now in the next section we'll have a look at where the dot product might be used

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in practice.

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So have a practice around create some matrices transpose them see if you can do a dot product and see

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in the next video.