show-how-the-axioms-for-a-vector-space-v-can-be-used-to-prove-the-elementary-properties-described-4

Question

Show how the axioms for a vector space $V$ can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom $2,$ Axioms 4 and 5 imply, respectively, that $\mathbf{0}+\mathbf{u}=\mathbf{u}$ and $-\mathbf{u}+\mathbf{u}=\mathbf{0}$ for all $\mathbf{u}$.
Fill in the missing axiom numbers in the following proof that $c \mathbf{0}=\mathbf{0}$ for every scalar $c$ .
$\begin{aligned} c \mathbf{0} &=c(\mathbf{0}+\mathbf{0}) \ &=c \mathbf{0}+c \mathbf{0} \end{aligned}$
by Axiom _____ (a)
by Axiom _____ (b)

Viswesh Uppalapati · Accepted Answer

in this video song, Call number 28 of Section four. Bone, which gives us two steps of a proven tells us what axioms. Uh, these two steps are possible by and the first one&#x27;s asked him for, as it says, that any vector is added to the zero vector. So any vector you in, uh, vector space would be equal to you plus zero or vice versa. And you is also here. Zero is the vector that is being attitude. So you can also be zero in this. Um, in this case, so zero vector. It also applies for the zero vector. If you were to be zero vector. So that&#x27;s why we&#x27;re were able to weaken. Read Dr Step One and Step two is given to us by axiom seven, which states that, uh, any scaler value see times two vectors u plus V would equal to see you plus CV and they&#x27;re interchangeable. So the scaler value just distributes to you to the Becker&#x27;s

Viswesh Uppalapati · Answer

in this video was selling problem number 26 of 2nd 4.1 it&#x27;s just basically asking us to complete a proof and list the axioms from the book that are used to complete this. Complete this proof. So this proves pretty simple. It just tells us that you plus negative u equals, um zero for any unique you in a vector space. Um, which is one of the given axioms. So So it tells us approve, um, or complete the proof where you plus w also equal zero. And we start by adding negative you to both sides. So this is what we get. Um, so next we would just group negative you. Plus you plus w equals negative. You plus zero or zero minus you something. Um, so this is stated an axiom Three for the actions you can look in. Look in the book. Um, where they give a definition of Ah, subspace, our vector space. So next we just simplify this thio. So we know that negative. You close you zero from one of the axioms and zero plus w equals negative. You plus zero. Bye axiom. Five Well and lastly, zero positive used his w and negative. You plus zero is just negative. You This is stated an axiom for So it&#x27;s a pretty simple proof. So we were just given. We&#x27;re just ask the company the proof.

Donald Albin · Answer

So I need to prove that you the inner product of you with V plus double you is the inner product of U and V plus the inner product of you and W. Okay, so this is very much like, um, Property three, which says that, um So I&#x27;m gonna write property three. I am going to write V plus W comma. You equals you w plus V w. Okay, well, this is the same as you, V plus W because of property, too. So Property three is here, okay? And one property, too is here. And then we can use just, um this is supposed to say you up here. Oh, my goodness is supposed to say that&#x27;s re do this. I think that is very wrong. Um, you double you. All right? I wrote this completely wrong. It&#x27;s supposed to save you and w you all right. So let&#x27;s let&#x27;s keep going. Um, Then we can use property to to just switch these two around. So that&#x27;s going to give us you the plus. You w Okay. So, again, you could write the&#x27;s just like last time in a different order, probably starting here and then writing that and then writing that and then writing that

Viswesh Uppalapati · Answer

in this video, we&#x27;re gonna, uh we&#x27;re asked us we&#x27;re gonna be solving problem number 30 of section 411 And the question basically asked us to prove that, um suppose c u equals zero for some non secular, non scaler are non zero scaler. See, And we have to show that you equals zero. And we have to mention any properties and axioms that we&#x27;ve used to solve this. So the first thing you wanna do is, um so here you is a vector. So you we want to add see times zero to both sides of the zero vector because, um, zero plus c times zero and this doesn&#x27;t really using the axiom were just adding something that both sides to change the to keep it close in the same. So so next. We wantto factor out the sea here, see times you plus zero equals, uh, the right side&#x27;s zero plus c zero, and this is possible by axiom. Uh, Axiom seven, which is basically the distributive property of vectors. Uh, which states that c times you any vector you plus me equals C u plus cv. So here we see that See you, um, equals zero it holds and then see time. Zero. There&#x27;s a term for a seat, I&#x27;m sure. Alexis next to the one. Ah, basically simplify this. So we just distributed out against he See you plus C. Times zero is just zero. So, yeah, see you. Sometimes you equals See Time&#x27;s zero by axiom four. And all that&#x27;s left to do is divide dividing both sides by cease seethe sees cancel so new equals zero and this practice will saw.

Donald Albin · Answer

I&#x27;m going to prove that the inner product of V and the zero vector is zero. Well, first of all, the zero vector is the same as zero times any vector. And so I can write that one vector times. Another vector is the same as this vector times. This vector that is using, um, property, too. We can reverse the order. Um, then using property three using property three. This is zero times you V, which is zero. And this is V zero. And so if you write this, you might want to start here and proceed in this direction. So what I see here is just using a definition. Then using this equals that that&#x27;s property to this equals that this that&#x27;s gonna be property three and then zero times anything is zero.

Linh Vu · Answer

were given invented Yuriko j B Therefore, we can together minus better your echo. Gee, the money I honest be and everybody line And we put the end up here. So your plus minus you it Would you go? Do I bless Manus Pain being plus minus b and then we get equal. Judah&#x27;s those zero and examine ico. Gina is rector here. Doesn&#x27;t means that you&#x27;re blessed, man. You echoed you deserve vector.

Problem

Show how the axioms for a vector space $V$ can be…

Problem 28 Medium Difficulty

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications