Love Pavel's work. A truly dedicated explainer of things! Taking a break from the tensor calculus to revisit the more elementary (yet infinitely powerful) Linear Algebra. Who woulda thought adding and multiplying could be so interesting?
These are wonderful videos so far. I took linear algebra back in university, but didn't pay enough attention to the course (maybe because I had no idea why I would want to learn how to add rows and columns of numbers together). Already I have a much better idea of why I would want to do this! I'm watching the rest of the series on Lemma, where I'm hoping the addition of problems will help me confirm that I'm understanding what you're teaching.
Yeah, mathematicians are known for their tendency to hide what's really exciting in their job from the outsiders :q They want to look like edgelords ;)
For the polynomial q. The two polynomials having root 1 can be written as (x-1)(ax+ b) and (x-1)(cx +d) now a linear combination would be (x-1)(e(ax+b) + g(cx+d)). Now the span of 2 polynomials of degree one is a plane, as is the span of the two polynomials you gave on the board. Therefore yes two quadratic polynomials having root one that are not multiples of each other will span the space of all quadratics having root 1.
I think that a simpler explanation for why those two polynomials at 6:03 cannot span the entire space of order 3 polynomials is because any linear combination of two even numbers (the final digits 8 and 2) can only result in an even digit. So, for starters, these two polynomials could never generate a polynomial ending in an odd number.
A simple explanation for the property of polynomial coefficients is that the sum of the coefficients is just the value of the function at one, so if the sum is zero, one is a root.
If "a", "b" and "c", are three Vectors but lie on the same line, then Span(a,b,c) is a single line (even though they are 3 unique Vectors). If "a", "b" and "c", are three Vectors but lie on the same Plane, then Span(a,b,c) is the Plan they lie (even though they are 3 unique Vectors). Only if "a", "b" and "c", do NOT lie on the same plane then Span (a,b,c) is the whole 3-D Space. So what kind of Span the Vectors would create is based on their "arrangement".
Hope all your films can add subtitles later,because my native language isn't English. Need to know the precise word you say in case I misunderstand the lesson. Thank you,Sir.
How to visualize the span of polynomials as effectively as geometric vectors? Does it mean that the polynomials created by linear combination can have roots all over x-axis? Or is it that if we plot polynomials created by linear combination, they fill complete graph paper? Why have not you used visuals for that? I am having a hard time understanding the role of coefficients in it. Please help.
The beauty of Linear Algebra is that it swing like a pendulum between geometric and algebraic worlds. Polynomials are an example from the algebraic world so you should often let go of you geometric intuition when thinking about polynomials and contemplate their algebraic properties.
Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.
Cracked a smile too, when you almost dropped your vectors :) Enjoying your videos so far, thank you!
Love Pavel's work. A truly dedicated explainer of things! Taking a break from the tensor calculus to revisit the more elementary (yet infinitely powerful) Linear Algebra. Who woulda thought adding and multiplying could be so interesting?
These are wonderful videos so far. I took linear algebra back in university, but didn't pay enough attention to the course (maybe because I had no idea why I would want to learn how to add rows and columns of numbers together).
Already I have a much better idea of why I would want to do this! I'm watching the rest of the series on Lemma, where I'm hoping the addition of problems will help me confirm that I'm understanding what you're teaching.
Yeah, mathematicians are known for their tendency to hide what's really exciting in their job from the outsiders :q They want to look like edgelords ;)
0:16 : Spanning set definition
0:26 : Geometric Vectors
2:58 : Polynomials
6:53 : Question
8:37 : Rn
For the polynomial q. The two polynomials having root 1 can be written as (x-1)(ax+ b) and (x-1)(cx +d) now a linear combination would be (x-1)(e(ax+b) + g(cx+d)). Now the span of 2 polynomials of degree one is a plane, as is the span of the two polynomials you gave on the board. Therefore yes two quadratic polynomials having root one that are not multiples of each other will span the space of all quadratics having root 1.
nicely observed.
I think that a simpler explanation for why those two polynomials at 6:03 cannot span the entire space of order 3 polynomials is because any linear combination of two even numbers (the final digits 8 and 2) can only result in an even digit. So, for starters, these two polynomials could never generate a polynomial ending in an odd number.
I'm really glad you made this (incorrect) argument. Don't forget that multiplication by ALL real numbers is allowed including 1/2, 1/4, etc.
A simple explanation for the property of polynomial coefficients is that the sum of the coefficients is just the value of the function at one, so if the sum is zero, one is a root.
+GG Nore See ruclips.net/video/azakOh7TMwo/видео.html
Maths is fun with your explanations sir
Excellent explanation!
If "a", "b" and "c", are three Vectors but lie on the same line, then Span(a,b,c) is a single line (even though they are 3 unique Vectors).
If "a", "b" and "c", are three Vectors but lie on the same Plane, then Span(a,b,c) is the Plan they lie (even though they are 3 unique Vectors).
Only if "a", "b" and "c", do NOT lie on the same plane then Span (a,b,c) is the whole 3-D Space.
So what kind of Span the Vectors would create is based on their "arrangement".
Great job.
Hope all your films can add subtitles later,because my native language isn't English.
Need to know the precise word you say in case I misunderstand the lesson.
Thank you,Sir.
How to visualize the span of polynomials as effectively as geometric vectors? Does it mean that the polynomials created by linear combination can have roots all over x-axis? Or is it that if we plot polynomials created by linear combination, they fill complete graph paper?
Why have not you used visuals for that? I am having a hard time understanding the role of coefficients in it. Please help.
The beauty of Linear Algebra is that it swing like a pendulum between geometric and algebraic worlds. Polynomials are an example from the algebraic world so you should often let go of you geometric intuition when thinking about polynomials and contemplate their algebraic properties.
excellent explanation
thank you sir
8:21 when you multiply by -1/3**
I've never seen anyone use sticks to model vector problems. Beats 3D drawings on 2D blackboards. Yeh we should be modelling 3D in 3D!
Sticks and stones!