Matrices, matrix multiplication and linear transformations | Linear algebra makes sense

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Matrices are often presented as a useful bookkeeping/ commutation tools to students- but there’s much more to them. When you understand what a Matrix really is so many parts of Linear Algebra will be completely obvious to you… including the formula for matrix multiplication and the fact that matrices don’t commute. So here's the big secret: A matrix is a linear transformation that eats a vectors and outputs another vector.

Homework questions:
Not all sized matrices can be multiplied together. Think about it in terms of them representing transformations from one space to another, and figure out which size matrices can be multiplied and explain why in the comments.

Music: Epidemic sound, Summer nights 2

This video is an Introduction to Matrices but could be useful revision for school/university. If you have an exam, good luck!

[ScienceAsylum]

8:59 "no. I don't expect anything anymore" ...this is likely the best plan for understanding quantum mechanics.

[e.s.r]

I find it helpful to think of matrix notation like functions. g(f(x)) would say, do f(x) first then do g to the result, so ABv would be like A(B(v)) with the little nested brackets, "Do B to the vector first then A to the result". Then it's clearer they don't commute, the wording is just confusing.

[TheViolaBuddy]

I do have to say that AB != BA is not quite as obvious as you make it seem. Especially since we just talked about linear transformations whose definition involves doing transformations in different orders and getting the same thing. But of course many lin. alg. classes don't go into the intuition of why it's not associative, which is what you were presenting here. Speaking of intuition, I guess it really is true that all matrices are transformations - at least all that I can think of. I knew that they _can_ be, but I never really thought of that as being their fundamental identity of a matrix.

As for the homework questions, they're less proof-y this week! I answered the multiple choice questions, but as for the first one, "When can you multiply an (n x m) matrix by a (p x q) one?" the answer is of course "m = p": the p x q matrix takes in a q-dimensional vector and spits out a p-dimensional one. You now need a matrix that can take in a p-dimensional vector (regardless of what it spits out), so our second (n x m) matrix has to accept a p-dimensional vector; that is, m = p.

[adityakhanna]

You're not lazy, you're resourceful!

[seanziewonzie]

>I don't understand why some courses insist on making linear algebra awful and algorithmic. What's the point of 'knowing' how to multiply matrices if you don't understand it.

From my experience, it's because a lot of students will whinge if you try and explain what things "mean". Professors often get stuck in a "you can't win with everyone" situation and decide to (or are instructed to) err towards teaching algorithms so we can at the very least guarantee that every engineer who passes the class knows how to multiply matrices. I disagree with this practice in the modern era -- MatLab exists, so the only challenge left is knowing which matrices to choose, and understanding the meaning of matrices is the only way to make this process straightforward!

Some schools have made the decision to make TWO "intro to linear algebra" courses. To anyone reading this who hopes to go into physics, I advise you to go into the more mature of the two. It is not harder, or more work; it is designed with a different ethos. The meanings of the objects of linear algebra are explained as well as they are in this series. Key words to look out for in the course descriptions are "proof-based" or "abstract vector spaces". There is probably no better example of the conceptual power of abstract vector spaces than states in QM.

[adarshkishore]

Your way of teaching and sense of humour is great!

[lare]

I think you can only multiply two matrices together if the output dimension (height) of the first matrix is equal to the input dimension (width) of the second. Otherwise it makes no sense, because you can't transform an n-dimensional vector space with a matrix meant to transform m-dimensional vector spaces where n != m.

[ThePCxbox]

You're literally better than khan academy and 3blue(if you know who that is) combined! Please never stop making videos, they dont even need to be animated

[viniciuslambardozzi]

Is this actually a recording of your hand as your write? It's so smooth, almost seems like a tool for animating over strokes.

[shivChitinous]

So there are constraints on which vectors can be acted upon by a transformation. You can’t apply a matrix with 6 columns on a vector with 3 bases, because each column basically tells you what to do to each of the basis vectors. And clearly a matrix spits out a vector that has the same no. of bases as the no. of rows. So an mxn matrix will give you a vector with m bases. And this vector can only be operated on by a pxq matrix B if q=m.

[GauravGandhiOfficial]

Why isn't the determinant of a 3×2 matrix defined? The input space is a 2D plane and output space is also a 2D plane in 3 dimensions. So our unit square in the input space has converted into a parallelogram on that plane in 3 dimensions. Thinking about it this way, shouldn't the determinant be defined?

[avanishpadmakar]

The
question at 7:26 were we supposed to assume the basis vectors were [1, 0], [0, 1] .Would in general for basis vectors [a, b] and [c, d]
M=a 0
b 0 ?

[louiebafford]

Amazing video the intuitive explanation was so well done

[abhishekbhattacharjee]

and yes we can also enjoy any research type lecture (with minimal animation if u want)...of course it'll consume lot of ur time but i think quality videos will enhance ideas for both...

[OnTheThirdDay]

Hello. Thank you for the videos.
Question: Should we will in the survey/poll questions if we are already familiar with Linear Algebra? (I am and did....)

[Hecatonicosachoron]

I'm surprised that there are no row operations, haha!
Interesting vids, it's always nice to see different presentations on the basics :)

[dylanparker]

wait, in that first poll question, i thought you were asking about a matrix that effectively made a dot product style projection. or even just taking the cosine of the first vector's angle in obtaining the 2nd vector?

that's non-linear, surely?

[Pyriphlegeton]

Is that AB=/=BA Thumbnail a psychological Trick to garner views from people hyped for Mamma Mia 2?

[patrickwienhoft]

3:14
Really small criticism: Try to make your voiceover order match what you're writing. You say "first do a lin. comb. and then multiply by 3" but at the same time you write the 3 first, then do the lin. comb. This can really throw some people off.
I imagine you do the visuals first an then do the audio? Do you have the audio written out when recording the visuals or do you only have a general outline but no conrete sentences?

[joaquinbadillogranillo]

A vector that's perpendicular to the projection line would become null when projected, then rotating it would leave it null. However if this is in R^2 rotating it would make it have a component with respect to the projection line and when projected it won't be null :)