Linear Algebra 2L: The Zero Vector - Does Nothing, Yet so Important!

[stendall]

Someone else may have said this, but it's also important to pay close attention to the zero vector, because some linear algebra operations, like dot products or inner products, can give scalar answers from vector inputs, so distinguishing the answer as the zero scalar or zero vector is very important for understanding the expression.

[MathTheBeautiful]

@Alana I'm glad you asked this question and I very much hope that my response is helpful.

There are two distinct mathematical disciplines with very different personalities and very little psychological overlap: 1. Linear Algebra the Beautiful and 2. Linear Algebra the Formal. This course covers the former and your question is from latter.

It is beyond obvious for any linear space considered in this course that there is only one element that acts as the zero vector. It is so obvious that one wouldn't even think of asking the question. Would one ever ask: how do you show that there is only one polynomial that equals x^2? Even if one *would* ask a question like that, I would answer by saying that there are a thousand other questions that are much more exciting to talk about. Linear Algebra the Beautiful is about moving forward and answering exciting questions with unknown answers.

Linear Algebra the Formal is about movement towards depth, rather than forward, and exploring the logical consistency of the foundations of the subject. It is an important discipline but, in my opinion, should only be studied by a handful of people and only after studying Linear Algebra the Beautiful - not before and not at the same time. So if you are trying to study both at the same time, please reconsider, and join us on our Linear Algebra the Beautiful journey.

Now to answer your formal question: if there are two zeros, 01 and 02, then consider the sum 01+02. On the one hand, it equals 01 and, on the other, 02 - so the two must be the same.

[PyMoondra]

Love your course. Didn't finish it last time, so going to finish it while along side a textbook.

[nimrod]

a very important and interesting point made explicit here. thanks for sharing this!

[manishbhagat]

Ok sir...very clear... but what a zero vector does to the vector space ....why it's so important for the very existence of the vector space....

[JimPIckins]

this guy is a genius. i look forward to watching these videos. they make sense. i wish he had a whiteboard though, or a non-chalk blackboard. i cringe every time he goes to use the chalk from the noise.

[antonellomascarello]

0:13 : Introducing THE zero vector
0:45 : THE zero vector in the space of geometric vectors
2:57 : ... In the space of polynomials
3:47 : ... In Rn
5:05 / 7:32 : IMPORTANT (pay attention referring to the zero vector)
7:39 : Example 1
9:13 : Example 2

[anthonypascual]

We should just denote all kinds of vectors with an arrow on the top. No BS, you know? We already started throwing triangles and shapes into math, what's an extra arrow to all that notation

[theporcupine]

wauw if it's that easy. thank you sir.
in my book there is just a confusing definition.

[cometmace]

8:19: "On the left we have a linear combination of three polynomials.."     How is (x+1)(x-1), in that form, a polynomial?  It looks like the product of two polynomials.

[thomasolson]

I have a vector in 3d. It has a square root of three and imaginary values mixed in. When I check its magnitude, it's fine. It works like any other ordinary vector. When I put the magnitude in the denominator all its values algebraically cancel when I rationalize the denominator and it's left with a zero vector. The vector has a magnitude of 1 if I simply square it immediately but it obviously has a magnitude of 0 if I simplify the whole thing before squaring each of the components.

The vector was designed to be a zero vector. I manipulated rotations in 3d with trinomials. I can't remember exactly what I did. I would show you the vector but I don't want to distract you from the conundrum.

[MathCuriousity]

Bro I appreciate your video but if this is a reflection of the depth you will cover, I am a bit worried; how can you talk about zero vector and not discuss additive identities as well as how we define addition will change the zero vector.

Hello!
Talking about the zero vector... How can I prove that there is only one zero vector in a vector space?

[entropyz]

@MathTheBeautiful I have a doubt that professors have not been able to answer. When we say the zero vector in a vector space, must that zero vector be composed of only zeros or can it be any number as long as it satisfies the given axioms and rules? Thank you for your answer.

[chrisarieloro]

Is (x-1)*(x+1) a linear combination? I thought it was just multiplication by numbers that counted as linear combinations.

[mikeschlarbaum]

wondering if you made a mistake in your video here; suppose I denote p(x) = x-1, q(x) = x+1, w(x) = x^2 - 1: then your linear combination of three vectors as stated in the video is equaivalent to taking p(x)q(x) - w(x), however linear combinations do not involve direct multiplication of vectors themselves, only scalar multiplication as defined in their respective field. Am I missing somehting?

[pratikghatge]

Hi Pavel, at 8:18 do you mean it is the linear combination of 2 polynomials or 3 polynomials? Thanks

[rahulthankachan]

Can Vectors be multiplied? You had mentioned in linear algebra we consider only sum of vectors and multiplication by scalar. I see X^2 in the video. I know this could be stupid doubt but please let me know. Thanks