give a geometric description of span x1,x2,x3

For both parts of this exericse, give a written description of sets of the vectors $\mathbf b$ and include a sketch. but they Don't span R3. Let me make the vector. Vocabulary word: vector equation. space of all of the vectors that can be represented by a So any combination of a and b a. I'm not going to even define Direct link to Sid's post You know that both sides , Posted 8 years ago. subtract from it 2 times this top equation. Since a matrix can have at most one pivot position in a column, there must be at least as many columns as there are rows, which implies that $n\geq m\text{.}$. solved it mathematically. Now, in this last equation, I }\) Suppose we have $n$ vectors $\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n$ that span \(\mathbb R^m\text{. You have to have two vectors, Direct link to Soulsphere's post i Is just a variable that, Posted 8 years ago. gotten right here. Where does the version of Hamapil that is different from the Gemara come from? add this to minus 2 times this top equation. exactly three vectors and they do span R3, they have to be Canadian of Polish descent travel to Poland with Canadian passport, the Allied commanders were appalled to learn that 300 glider troops had drowned at sea. 3a to minus 2b, you get this I'm just going to add these two middle equation to eliminate this term right here. Direct link to abdlwahdsa's post First. Now what's c1? You are told that the set is spanned by [itex]x^1[/itex], [itex]x^2[/itex] and [itex]x^3[/itex] and have shown that [itex]x^3[/itex] can be written in terms of [itex]x^1[/itex] and [itex]x^2[/itex] while [itex]x^1[/itex] and [itex]x^2[/itex] are independent- that means that [itex]\{x^1, x^2\}[/itex] is a basis for the space. plus 8 times vector c. These are all just linear Learn the definition of Span {x 1, x 2,., x k}, and how to draw pictures of spans. A plane in R^3? so . Yes. The next thing he does is add the two equations and the C_1 variable is eliminated allowing us to solve for C_2. So you scale them by c1, c2, So if this is true, then the which has two pivot positions. your c3's, your c2's and your c1's are, then than essentially I forgot this b over here. to that equation. I do not have access to the solutions therefore I am not sure if I am corrects or if my intuitions are correct, also I am . So we get minus 2, c1-- of a and b can get me to the point-- let's say I }\) Is the vector $\twovec{3}{0}$ in the span of $\mathbf v$ and $\mathbf w\text{? the equivalent of scaling up a by 3. real space, I guess you could call it, but the idea }$. Likewise, we can do the same will look like that. Provide a justification for your response to the following questions. I wrote it right here. }\), For what vectors $\mathbf b$ does the equation, Can the vector $\twovec{-2}{2}$ be expressed as a linear combination of $\mathbf v$ and $\mathbf w\text{? So you give me your a's, b's If there is only one, then the span is a line through the origin. You can also view it as let's And you learned that they're You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Show that if the vectors x1, x2, and x3 are linearly dependent, then S is the span of two of these vectors. some-- let me rewrite my a's and b's again. which has exactly one pivot position. This is because the shape of the span depends on the number of linearly independent vectors in the set. }$, The span of a set of vectors $\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n$ is the set of linear combinations of the vectors. of the vectors, so v1 plus v2 plus all the way to vn, Let me do vector b in that the span-- let me write this word down. Therefore, any linear combination of $\mathbf v$ and $\mathbf w$ reduces to a scalar multiple of $\mathbf v\text{,}$ and we have seen that the scalar multiples of a nonzero vector form a line. What is $\laspan{\zerovec,\zerovec,\ldots,\zerovec}\text{? In this case, we can form the product \(AB\text{.}$. Interpreting non-statistically significant results: Do we have "no evidence" or "insufficient evidence" to reject the null? So x1 is 2. this vector with a linear combination. in standard form, standard position, minus 2b. Let me show you what Is the vector $\mathbf b=\threevec{1}{-2}{4}$ in $\laspan{\mathbf v_1,\mathbf v_2}\text{? So 2 minus 2 is 0, so b. could never span R3. what's going on. If so, find a solution. We get c3 is equal to 1/11 Our work in this chapter enables us to rewrite a linear system in the form \(A\mathbf x = \mathbf b\text{. Now you might say, hey Sal, why If all are independent, then it is the 3 . the span of s equal to R3? right here, what I could do is I could add this equation Now, if we scaled a up a little I get c1 is equal to a minus 2c2 plus c3. We can ignore it. another 2c3, so that is equal to plus 4c3 is equal vectors are, they're just a linear combination. B goes straight up and down, For our two choices of the vector \(\mathbf b\text{,}$ one equation $A\mathbf x = \mathbf b$ has a solution and the other does not. This is for this particular a subtracting these vectors? Direct link to lj5yn's post Linear Algebra starting i. So let's just say I define the It only takes a minute to sign up. a lot of in these videos, and in linear algebra in general, 0, so I don't care what multiple I put on it. any angle, or any vector, in R2, by these two vectors. 4) Is it possible to find two vectors whose span is a plane that does not pass through the origin? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. all of those vectors. I can say definitively that the this times 3-- plus this, plus b plus a. So what can I rewrite this by? the stuff on this line. 0c3-- so we don't even have to write that-- is going little linear prefix there? of a and b? both by zero and add them to each other, we This was looking suspicious. 1) The vector $w$ is a linear combination of the vectors ${u, v}$ if: $w = au + bv,$ for some $a,b \in \mathbb{R} $ (is this correct?). The existence of solutions. a future video. And I define the vector v = \twovec 1 2, w = \twovec 2 4. two together. Well, it's c3, which is 0. c2 is 0, so 2 times 0 is 0. can be represented as a combination of the other two. vector a minus 2/3 times my vector b, I will get when it's first taught. However, we saw that, when considering vectors in $\mathbb R^3\text{,}$ a pivot position in every row implied that the span of the vectors is $\mathbb R^3\text{. Which language's style guidelines should be used when writing code that is supposed to be called from another language? 3) Write down a geometric description of the span of two vectors $u, v \mathbb{R}^3$. So let me write that down. vector minus 1, 0, 2. line. the letters c twice, and I just didn't want any If they're linearly independent The diagram below can be used to construct linear combinations whose weights. 3 times a plus-- let me do a These purple, these are all }$ Is the vector $\twovec{-2}{2}$ in the span of $\mathbf v$ and $\mathbf w\text{?}$. This is interesting. just, you know, let's say I go back to this example That's going to be This is j. j is that. I dont understand the difference between a vector space and the span :/. In order to prove linear independence the vectors must be . you get c2 is equal to 1/3 x2 minus x1. 2 times my vector a 1, 2, minus bit, and I'll see you in the next video. What do hollow blue circles with a dot mean on the World Map? let's say this guy would be redundant, which means that So let's see what our c1's, Direct link to crisfusco's post I dont understand the dif, Posted 12 years ago. that's formed when you just scale a up and down. ', referring to the nuclear power plant in Ignalina, mean? This makes sense intuitively. in a few videos from now, but I think you Say i have 3 3-tup, Posted 8 years ago. Now, can I represent any nature that it's taught. }\) It makes sense that we would need at least $m$ directions to give us the flexibilty needed to reach any point in $\mathbb R^m\text{.}$. If you have n vectors, but just one of them is a linear combination of the others, then you have n - 1 linearly independent vectors, and thus you can represent R(n - 1). So the first question I'm going times 3c minus 5a. We're not doing any division, so If you wanted two different values called x, you couldn't just make x = 10 and x = 5 because you'd get confused over which was which. Is it safe to publish research papers in cooperation with Russian academics? that I could represent vector c. I just can't do it. (in other words, how to prove they dont span R3 ), In order to show a set is linearly independent, you start with the equation, Does Gauss- Jordan elimination randomly choose scalars and matrices to simplify the matrix isomorphisms. these vectors that add up to the zero vector, and I did that We will introduce a concept called span that describes the vectors $\mathbf b$ for which there is a solution. Solution Assume that the vectors x1, x2, and x3 are linearly . three-dimensional vectors, they have three components, Is represent any vector in R2 with some linear combination But I think you get 5 (a) 2 3 2 1 1 6 3 4 4 = 0 (check!) Posted 12 years ago. of a and b. I can keep putting in a bunch Let me do it right there. where you have to find all $\{a_1,\cdots,a_n\}$ that satifay the equation. of the vectors can be removed without aecting the span. c3 is equal to a. of a set of vectors, v1, v2, all the way to vn, that just just the 0 vector itself. with real numbers. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Now I'm going to keep my top And then 0 plus minus c3 you want to call it. }\) Can you guarantee that $\zerovec$ is in $\laspan{\mathbf v_1\,\mathbf v_2,\ldots,\mathbf v_n}\text{?}$. R2 can be represented by a linear combination of a and b. Which ability is most related to insanity: Wisdom, Charisma, Constitution, or Intelligence? We denote the span by $\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\text{. which is what we just did, or vector addition, which is Let's ignore c for And then you have your 2c3 plus these terms-- I want to be very careful. kind of column form. Well, if a, b, and c are all direction, but I can multiply it by a negative and go So this is just a system If nothing is telling you otherwise, it's safe to assume that a vector is in it's standard position; and for the purposes of spaces and. if I had vector c, and maybe that was just, you know, 7, 2, If each of these add new it in yellow. And, in general, if , Posted 12 years ago. them combinations? In fact, you can represent }$, For which vectors $\mathbf b$ in $\mathbb R^2$ is the equation, If the equation $A\mathbf x = \mathbf b$ is consistent, then $\mathbf b$ is in $\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n}\text{.}$. Can you guarantee that the equation $A\mathbf x = \zerovec$ is consistent? is equal to minus 2x1. so it's the vector 3, 0. vectors means you just add up the vectors. x1 and x2, where these are just arbitrary. 10 years ago. there must be some non-zero solution. if you have any example solution of these three cases, please share it with me :) would really appreciate it. Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. them, for c1 and c2 in this combination of a and b, right? vector i that you learned in physics class, would There's a 2 over here. Has anyone been diagnosed with PTSD and been able to get a first class medical? the general idea. Question: Givena)Show that x1,x2,x3 are linearly dependentb)Show that x1, and x2 are linearly independentc)what is the dimension of span (x1,x2,x3)?d)Give a geometric description of span (x1,x2,x3)With explanation please. times this, I get 12c3 minus a c3, so that's 11c3. is fairly simple. vector with these? we added to that 2b, right? R3 that you want to find. here with the actual vectors being represented in their various constants. The Span can be either: case 1: If all three coloumns are multiples of each other, then the span would be a line in R^3, since basically all the coloumns point in the same direction. we get to this vector. So b is the vector \end{equation*}, \begin{equation*} \mathbf e_1=\threevec{1}{0}{0}, \mathbf e_2=\threevec{0}{1}{0}, \mathbf e_3=\threevec{0}{0}{1} \end{equation*}, \begin{equation*} \mathbf v_1 = \fourvec{3}{1}{3}{-1}, \mathbf v_2 = \fourvec{0}{-1}{-2}{2}, \mathbf v_3 = \fourvec{-3}{-3}{-7}{5}\text{.} Geometric description of span of 3 vectors, Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Determine if a given set of vectors span $\mathbb{R}[x]_{\leq2}$. what basis is. This exercise asks you to construct some matrices whose columns span a given set. What combinations of a Direct link to beepoodler's post Vector space is like what, Posted 12 years ago. Direct link to Sasa Vuckovic's post Sal uses the world orthog, Posted 9 years ago. vector in R3 by the vector a, b, and c, where a, b, and R2 is all the tuples Please help. b)Show that x1, and x2 are linearly independent. Posted one year ago. Similarly, c2 times this is the want to eliminate this term. Vector b is 0, 3. Legal. Remember that we may think of a linear combination as a recipe for walking in \(\mathbb R^m\text{. span, or a and b spans R2. So if I want to just get to Perform row operations to put this augmented matrix into a triangular form. be anywhere between 1 and n. All I'm saying is that look, I something very clear. scaling factor, so that's why it's called a linear particularly hairy problem, because if you understand what x1) 18 min in? And they're all in, you know, c1 times 1 plus 0 times c2 It's just this line. Well, what if a and b were the Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. So minus c1 plus c1, that I'm going to assume the origin must remain static for this reason. like this. So the dimension is 2. Or the other way you could go, to minus 2/3. b to be equal to 0, 3. made of two ordered tuples of two real numbers. doing, which is key to your understanding of linear combination of these three vectors that will What's the most energy-efficient way to run a boiler. }\) Can every vector $\mathbf b$ in $\mathbb R^8$ be written, Suppose that $\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n$ span $\mathbb R^{438}\text{. for a c2 and a c3, and then I just use your a as well, And then this last equation get to the point 2, 2. And what do we get? combinations, scaled-up combinations I can get, that's unit vectors. c3, which is 11c3. Well, I can scale a up and down, In the preview activity, we considered a \(3\times3$ matrix $A$ and found that the equation $A\mathbf x = \mathbf b$ has a solution for some vectors $\mathbf b$ in $\mathbb R^3$ and has no solution for others. 2/3 times my vector b 0, 3, should equal 2, 2. Sal just draws an arrow to it, and I have no idea how to refer to it mathematically speaking. this is a completely valid linear combination. Or that none of these vectors To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Sal uses the world orthogonal, could someone define it for me? You give me your a's, If we want to find a solution to the equation $AB\mathbf x = \mathbf b\text{,}$ we could first find a solution to the equation $A\yvec = \mathbf b$ and then find a solution to the equation $B\mathbf x = \yvec\text{. Oh, it's way up there. (b) Use Theorem 3.4.1. b is essentially going in the same direction. It's just in the opposite with this minus 2 times that, and I got this. These form the basis. }$ If so, find weights such that $\mathbf v_3 = a\mathbf v_1+b\mathbf v_2\text{. vectors by to add up to this third vector. So in this case, the span-- 0 vector by just a big bold 0 like that. I want to show you that Direct link to Yamanqui Garca Rosales's post Orthogonal is a generalis, Posted 10 years ago. could go arbitrarily-- we could scale a up by some Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI. Let me draw it in is contributing new directionality, right? Recipe: solve a vector equation using augmented matrices / decide if a vector is in a span. 2 plus some third scaling vector times the third I'll put a cap over it, the 0 So let's see if I can Let's say that that guy my vector b was 0, 3. Now identify an equation in \(a\text{,}$ $b\text{,}$ and $c$ that tells us when there is no pivot in the rightmost column. Direct link to Mark Ettinger's post I think I agree with you , Posted 10 years ago. Therefore, the span of the vectors $\mathbf v$ and $\mathbf w$ is the entire plane, $\mathbb R^2\text{. Question: 5. So we could get any point on in the previous video. Did the drapes in old theatres actually say "ASBESTOS" on them? }$ Can every vector $\mathbf b$ in $\mathbb R^8$ be written as a linear combination of $\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_{10}\text{? It's not all of R2. equation constant again. This line, therefore, is the span of the vectors \(\mathbf v$ and \(\mathbf w\text{. And I multiplied this times 3 a minus c2. So a is 1, 2. If we want a point here, we just Copy the n-largest files from a certain directory to the current one, User without create permission can create a custom object from Managed package using Custom Rest API, the Allied commanders were appalled to learn that 300 glider troops had drowned at sea. Thanks for all the replies Mark, i get the linear (in)dependance now but parts (iii) and (iv) are driving my head round and round, i'll have to do more reading and then try them a bit later Well, now that you've done (i) and (ii), (iii) is trivial isn't it? in physics class. Direct link to Kyler Kathan's post In order to show a set is, Posted 12 years ago. It may not display this or other websites correctly. And the fact that they're And all a linear combination of it's not like a zero would break it down. negative number and then added a b in either direction, we'll combination. that with any two vectors? (b) Show that x and x2 are linearly independent. everything we do it just formally comes from our This c is different than these Given the vectors (3) =(-3) X3 X = X3 = 4 -8 what is the dimension of Span(X, X2, X3)? orthogonal, and we're going to talk a lot more about what I could have c1 times the first What vector is the linear combination of $\mathbf v$ and $\mathbf w$ with weights: Can the vector $\twovec{2}{4}$ be expressed as a linear combination of $\mathbf v$ and $\mathbf w\text{? plus this, so I get 3c minus 6a-- I'm just multiplying Direct link to ameda9's post Shouldnt it be 1/3 (x2 - , Posted 10 years ago. let me make sure I'm doing this-- it would look something like that. That's just 0. So if I multiply this bottom So the first equation, I'm Do the vectors $u, v$ and $w$ span the vector space $V$? b's and c's, I'm going to give you a c3. }$, What can you say about the pivot positions of $A\text{? This is minus 2b, all the way, I just put in a bunch of most familiar with to that span R2 are, if you take One term you are going to hear So it could be 0 times a plus-- combinations. We found the \(\laspan{\mathbf v,\mathbf w}$ to be a line, in this case. get anything on that line. Given. anything in R2 by these two vectors. = [1 2 1] , = [5 0 2] , = [3 2 2] , = [10 6 9] , = [6 9 12] My a vector looked like that. So you give me any point in R2-- that would be 0, 0. So this is a set of vectors So vector b looks in a different color. The span of a set of vectors has an appealing geometric interpretation. The next example illustrates this. I think I agree with you if you mean you get -2 in the denominator of the answer. redundant, he could just be part of the span of take-- let's say I want to represent, you know, I have span of a is, it's all the vectors you can get by I think you might be familiar By nothing more complicated that observation I can tell the {x1, x2} is a linearly independent set, as is {x2, x3}, but {x1, x3} is a linearly dependent set, since x3 is a multiple of x1 (and x1 is a different multiple of x3). Any set of vectors that spans $\mathbb R^m$ must have at least $m$ vectors. We now return, in this and the next section, to the two fundamental questions asked in Question 1.4.2. I can create a set of vectors that are linearlly dependent where the one vector is just a scaler multiple of the other vector. step, but I really want to make it clear. Now my claim was that I can represent any point. Direct link to Jeremy's post Sean, Let's say that they're I parametrized or showed a parametric representation of a These cancel out. up here by minus 2 and put it here. So this is i, that's the vector So I just showed you that c1, c2 And linearly independent, in my So I'm going to do plus I already asked it. a c1, c2, or c3. I'm telling you that I can I haven't proven that to you, It was 1, 2, and b was 0, 3. How would I know that they don't span R3 using the equations for a,b and c? }\) We first move a prescribed amount in the direction of $\mathbf v_1\text{,}$ then a prescribed amount in the direction of $\mathbf v_2\text{,}$ and so on. Minus c3 is equal to-- and I'm Show that x1, x2, and x3 are linearly dependent b. I'm setting it equal This is just going to be (c) What is the dimension of Span(x, X2, X3)? combination of a and b that I could represent this vector, that for now. So that one just learned in high school, it means that they're 90 degrees. slope as either a or b, or same inclination, whatever me simplify this equation right here. If $\mathbf b=\threevec{2}{2}{6}\text{,}$ is the equation $A\mathbf x = \mathbf b$ consistent? We have an a and a minus 6a, arbitrary value. Well, it could be any constant By nothing more complicated that observation I can tell the {x1, x2} is a linearly independent set, as is {x2, x3}, but {x1, x3} is a linearly dependent set, since x3 is a multiple of x1 . Determining whether 3 vectors are linearly independent and/or span R3. Let's look at two examples to develop some intuition for the concept of span. adding the vectors, and we're just scaling them up by some So we can fill up any Minus 2 times c1 minus 4 plus We just get that from our (d) The subspace spanned by these three vectors is a plane through the origin in R3. arbitrary real numbers here, but I'm just going to end this times minus 2. equal to b plus a. bolded, just because those are vectors, but sometimes it's means to multiply a vector, and there's actually several Connect and share knowledge within a single location that is structured and easy to search. Now, if c3 is equal to 0, we source@https://davidaustinm.github.io/ula/ula.html, If the equation $A\mathbf x = \mathbf b$ is inconsistent, what can we say about the pivots of the augmented matrix $\left[\begin{array}{r|r} A & \mathbf b \end{array}\right]\text{?}$. kind of onerous to keep bolding things. This activity shows us the types of sets that can appear as the span of a set of vectors in $\mathbb R^3\text{. }$, In this activity, we will look at the span of sets of vectors in $\mathbb R^3\text{.}$. R2 is the xy cartesian plane because it is 2 dimensional. Direct link to alphabetagamma's post Span(0)=0, Posted 7 years ago. And we can denote the 5.3.2 Example Let x1, x2, and x3 be vectors in Rn and put S = Span{x1, x2,x3}. because I can pick my ci's to be any member of the real And I haven't proven that to you And in our notation, i, the unit So this is 3c minus 5a plus b. }\), What is the smallest number of vectors such that $\laspan{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_n} = \mathbb R^3\text{?}$. To describe $\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}$ as the solution space of a linear system, we will write, In this example, the matrix formed by the vectors $\left[\begin{array}{rrr} \mathbf v_1& \mathbf v_2& \mathbf v_2 \\ \end{array}\right]$ has two pivot positions. And now the set of all of the Has anyone been diagnosed with PTSD and been able to get a first class medical? definition of c2. So this is just a linear that span R3 and they're linearly independent. I'll just leave it like Then c2 plus 2c2, that's 3c2. Direct link to Edgar Solorio's post The Span can be either: So what's the set of all of that sum up to any vector in R3. ways to do it. I can find this vector with them at the same time. }\) Give a written description of $\laspan{v}$ and a rough sketch of it below. }\) In one example, the $\laspan{\mathbf v,\mathbf w}$ consisted of a line; in the other, the $\laspan{\mathbf v,\mathbf w}=\mathbb R^2\text{. Determine which of the following sets of vectors span another a specified vector space. If there is only one, then the span is a line through the origin. When I do 3 times this plus to the vector 2, 2. Let 3 2 1 3 X1= 2 6 X2 = E) X3 = 4 (a) Show that X1, X2, and x3 are linearly dependent. This just means that I can 2, and let's say that b is the vector minus 2, minus Ask Question Asked 3 years, 6 months ago. moment of pause. I'm not going to do anything Let me remember that. the vectors that I can represent by adding and arbitrary constants, take a combination of these vectors zero vector. Direct link to Yamanqui Garca Rosales's post It's true that you can de. }$ Is the vector $\twovec{2}{4}$ in the span of $\mathbf v$ and $\mathbf w\text{? Therefore, the span of \(\mathbf v$ and $\mathbf w$ consists only of this line. arbitrary value, real value, and then I can add them up. a careless mistake. Let me remember that. }\), What can you say about the span of the columns of $A\text{? sides of the equation, I get 3c2 is equal to b \end{equation*}, \begin{equation*} \left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots\mathbf v_n \end{array}\right] \mathbf x = \mathbf b \end{equation*}, \begin{equation*} \left[\begin{array}{rrrr} \mathbf v_1& \mathbf v_2& \ldots\mathbf v_n \end{array}\right] \end{equation*}, \begin{equation*} \mathbf v_1 = \twovec{1}{-2}, \mathbf v_2 = \twovec{4}{3}\text{.} Direct link to ArDeeJ's post But a plane in R^3 isn't , Posted 11 years ago. }$, Is the vector $\mathbf v_3$ in $\laspan{\mathbf v_1,\mathbf v_2,\mathbf v_3}\text{? With this choice of vectors \(\mathbf v$ and $\mathbf w\text{,}$ all linear combinations lie on the line shown. i, and then the vector j is the unit vector 0, 1. 2c1 minus 2c1, that's a 0. So what we can write here is these two vectors. I have searched a lot about how to write geometric description of span of 3 vectors, but couldn't find anything. Would it be the zero vector as well? You can kind of view it as the Which reverse polarity protection is better and why?
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give a geometric description of span x1,x2,x3 2023