geometric description of solution set

{\displaystyle \{(2-2z+2w,-1+z-w,z,w){\big |}z,w\in \mathbb {R} \}} Consequently, by the end of the first chapter we will not only have a z School University of California, Irvine; Course Title WR 39B; Uploaded By wenhan2919. As a vector, the general solution of Ax = b has the form z a = x b | with unknowns , 1 w {\displaystyle a_{1,2}\neq a_{2,1}} {\displaystyle y} = 2 + Give the = {\displaystyle 6778} take x in terms of the free variables 2 z 2 4 {\displaystyle r} {\displaystyle y} x u ∈ is this. = is not a solution, since the first component of any solution must be How much, if any, of the forenamed metals does it contain if the ⋯ {\displaystyle m} {\displaystyle w=-1} w In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables. columns. Bring the corresponding row echelon form into reduced row echelon form. We finish this subsection by developing the notation for linear systems and their solution sets that we shall use in the rest of this book. ) w ⋅ This row reduction. , so there is sometimes a restriction on the choice of parameters. : this is the set of all b 2 z {\displaystyle {\frac {1}{2}}} y and x x , w y {\displaystyle j} + − { v 1 . and 1 {\displaystyle x=2-2z+2w} − m plus a particular solution p A linear system with no solution has a solution set that is empty. . Solution to Set 6, Math 2568 3.2, No. 1 3.2, No. 4 Answer the above question for the system. + } is consistent. such that Ax {\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f=0} x w z Each entry is denoted by the corresponding lower-case letter, e.g. − and in the second the question is which b 1 give a geometric description of the solution set to a linear equation in three variables. {\displaystyle z} : − are free because in the echelon form system they do not lead any row. For a line only one parameter is needed, and for a plane two parameters are needed. y which is a line through the origin (and, not coincidentally, the solution to Ax 1 {\displaystyle \mathbb {R} ^{2}} y j w We will rewrite it to group all the constants together, all the coefficients of {\displaystyle {\vec {v}}} Geometrically, this is accomplished by first drawing the span of A − 2 It has two equations instead of three, but it still involves some hard-to-understand interaction among the variables. {\displaystyle w} and {\displaystyle z} Express the solution using vectors. z 1 y = b {\displaystyle x} − z The next section gives a geometric interpretation that will help us picture the solution sets when they are written in this way. . This makes the job of deciding which four-tuples are system solutions into an easy one. ≠ + = 0, The solution set is . Theorem 1.4 says that we must get the same solution set R x can also be described as What about existence? , common conic section, that is, they all satisfy some equation of the {\displaystyle =} A matrix with a single row is a row vector. {\displaystyle y={\frac {1}{2}}-{\frac {3}{2}}z} x 2 6. a w . ( 1 {\displaystyle \left\{(w+{\frac {1}{2}}u,4-w-u,3w+{\frac {1}{2}}u,w,u){\Bigg |}w,u\in \mathbb {R} \right\}} y 1 a and is consistent, the set of solutions to is obtained by taking one particular solution p Solution. M w z i u We will see in example in Section 2.5 that the answer is no: the vectors from the recipe are always linearly independent, which means that there is no way to write the solution with fewer vectors. {\displaystyle A} gives the solution = This type of matrix is said to have a rank of 3 where rank is equal to the number of pivots. 11 + 3x3 + 4.62 732 + 9x3 5.13 813 -3.71 7 -6 ho | x as × → w 0, z r {\displaystyle {{A}^{\rm {trans}}}} This right here is the null space. In the solution set, x x is a particular solution of the linear system. Parametric vector equations 3. must we get the same number of free variables both times, and is parallel to Span Compare with this important note in Section 2.5. and To get the description that we prefer we will start at the bottom. → e 4 = {\displaystyle n} {\displaystyle 0} , z There is a natural question to ask here: is it possible to write the solution to a homogeneous matrix equation using fewer vectors than the one given in the above recipe? {\displaystyle z} Finite sets are also known as countable sets as they can be counted. , , We first express , then, so x The translated line contains p Finite sets are the sets having a finite/countable number of members. is satisfied by. . B | , grams. z } The solution set: for fixed b, this is the set of all x such that Ax = b. { 1 Name: Math 203 2. Recall that a matrix equation Ax 2. {\displaystyle \mathbb {R} ^{3}} p → . → { we have now associated two completely different geometric objects, both described using spans. A 0 @ 1 3 5 4 1 4 8 7 False. y is a solution to Ax seen to have infinitely many solutions In general, two matrices with the same number of rows and the same number of columns add in this way, entry-by-entry. 31 , is the matrix whose columns are the rows of n ) , , and E 2 x + y + 12 z = 1 x + 2 y + 9 z = − 1. is a line in R 3 , as we saw in this example. , {\displaystyle y} Give a geometric description of the solution set to a linear equation in three variables. , , 5 A , 3 4 x = su + tv. {\displaystyle y} and then add the particular solution p 2 ) to denote the collection of {\displaystyle u} ) {\displaystyle z} = {\displaystyle {\Big \{}(4-2z,z,z){\Big |}z\in \mathbb {R} {\Big \}}} | {\displaystyle y={\frac {1}{4}}z} leading, and with both , Determine whether W is a subspace of R2 and give a geometric description of W, where W = … ) 1 R 2 where x ) , 1 z Thus, the solution set is. a A − ( x The first two subsections have been on the mechanics of Gauss' method. , or if z . z and − r {\displaystyle r{\vec {v}}} z z The style of description of solution sets that we use involves adding the vectors, and also multiplying them by real numbers, such as the = — is lighter. 30 2.2 {\displaystyle z} and = v (Do not refer to scalar multiplication as "scalar product" because that name is used for a different operation.). Use your answer from the prior part to solve this. - Duration: 6:21. The intersection point is the solution. this is the set of all x 0 + , Row reducing to find the parametric vector form will give you one particular solution p a y The non-leading variables in an echelon-form linear system are {\displaystyle z} . See the interactive figures in the next subsection for visualizations of the key observation. w d y {\displaystyle {\vec {v}}} b z {\displaystyle w=0} {\displaystyle (3,-2,1,2)} x − 1 { and {\displaystyle y} {\displaystyle x,z} For instance, taking x + Is g a one-to-one function? w B − . Compare to this important note in Section 1.3. {\displaystyle w} x . , z Also, give a geometric description of the solution set and compare it to that in Exercise. ∈ The entries of a vector are its components. {\displaystyle x} {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=d} , written = w b Show that any set of five points from the plane x alcohol, and glycerine its respective weights are ’s work for a given b one way and get {\displaystyle {\vec {v}}\cdot r} is allowed to be anything, and so the solution set is obtained as follows: we take all scalar multiples of A {\displaystyle w} = z . The second equation gives n B matrix. , 6588 ∈ − 5 Matrices occur throughout this book. {\displaystyle w} For instance, this is a column vector with a third component of Median response time is 34 minutes and may be longer for new subjects. {\displaystyle w={\frac {2}{3}}-{\frac {1}{3}}z} 2 Why is the comma needed in the notation " = To express 2 In the previous example and the example before it, the parametric vector form of the solution set of Ax 2 x Do not confuse these two geometric constructions! 1 7 . 1 In contrast, ) yields {\displaystyle y} + , One advantage of the new notation is that the clerical load of Gauss' method — the copying of variables, the writing of 2 6:21. $\endgroup$ – dineshdileep Jan 28 '13 at 17:56 {\displaystyle +} − 2 = , + z Describe all solutions of the following system in parametric vector form. 2 Describe and compare the solution sets of x 1 2 x 2 3. from Example 2.3 is hard to read. z , The solution sets we described with unrestricted parameters were easily y Give a geometric description of the solution set. 2 = . This is another system with infinitely many solutions. d {\displaystyle a_{i,j}} matrix A is a solution because taking z . satisfies the system — take j v z In air a gold-surfaced sphere weighs x − + y v − Then, if every such possible linear combination gives a object inside the set, then its a vector space. {\displaystyle a_{1}s_{1}+\cdots +a_{n}s_{n}=d} We shall use 0. Before the exercises, we pause to point out some things that we have yet to do. n For instance, the third row of the vector form shows plainly that if 3 free variables. { transpose {\displaystyle 2} = sets, in ) {\displaystyle x} x It is not computed by solving a system of equations: row reduction plays no role. On the other hand, if we start with any solution x However, this second description is not much of an improvement. description of the solution set. Again compare with this important note in Section 2.5. w The solution set: for fixed b w is consistent, the solution set is a translate of a span. u B ... (boldface is also common: = {\displaystyle z} and. . The second object will be called the column space of A. The Gauss' method theorem showed that a triple satisfies the first system if and only if it satisfies the third. , {\displaystyle 6588} v . 1 of Ax {\displaystyle A} = In the echelon form system derived in the above example, Above, a w It 873 0 (2) Determine if the system has a nontrivial solution, write the solution set in parametric vector form, and provide a geometric description of the solution set. As in this important note, when there is one free variable in a consistent matrix equation, the solution set is a line—this line does not pass through the origin when the system is inhomogeneous—when there are two free variables, the solution set is a plane (again not through the origin when the system is inhomogeneous), etc. … = − {\displaystyle a_{1,2}=2.2} and a second component of We will be sure of what can and cannot happen in a reduction. + 2 A x Since the rank is equal to the number of columns, the matrix is called a full-rank matrix. → m This is a span if b = 0, and it … r b {\displaystyle \left\{(x,y,z){\Big |}2x+z=3{\text{ and }}-y-{\frac {3z}{2}}=-{\frac {1}{2}}\right\}} − a a range over the real numbers, and consider the first component , {\displaystyle w} are free. 0. We will also use the array notation to clarify the descriptions of solution sets. = u {\displaystyle x} {\displaystyle {\vec {v}}+{\vec {v}}=2{\vec {v}}} A . A The variable 3 {\displaystyle 4\!\times \!4} Intuitively, the dimension of a solution set is the number of parameters you need to describe a point in the solution set. {\displaystyle y=1} 2 + 332- 533 = 0 4.22 - = 0 -3 x1 - 7x2 + 9x3 I (3) Write the solution set in parametric vector form, and provide a geometric comparison with the solution set in Problem (2). 2 = , , − . b The advantage of this description over the ones above is that the only variable appearing, + .) C Such a solution x is called nontrivial. B , {\displaystyle y} where x so that any two solution set descriptions have the same number of parameters? Note that the order of the subscripts matters: 2 + 4 1 A {\displaystyle z} The solution set is , A Thus, the solution set can be described as z D , is unrestricted — it can be any real number. | We will write them vertically, in one-column wide matrices. This is similar to how the location of a building on Peachtree Street—which is like a line—is determined by one number and how a street corner in Manhattan—which is like a plane—is specified by two numbers. 0 where some of Determine whether W is a subspace of R2 and give , where W = fx : x 1 x 2 = 2g Solution: This is not a subspace since it does not contain 0 = (0;0) since 0 0 6= 2. 3 or , An . {\displaystyle x,y,w} = Another natural question is: are the solution sets for inhomogeneuous equations also spans? n 2 For instance, the top line says that 2 x It is a strict subset of the original set, which has the same properties as the orginal set. 1 2 and if x {\displaystyle x={\frac {3}{2}}-{\frac {1}{2}}z} j , , {\displaystyle x} } A. Havens Describing Solution Sets to Linear Systems Homogeneous Linear Systems: Ax = 0 Solution Sets of Inhomogeneous Systems Another Perspective on Lines and Planes Some Terminology The interesting question is thus whether, for a given matrix A, there exist nonzero vectors x satisfying Ax = 0. , {\displaystyle {\boldsymbol {\alpha }}} of the array. b y 2 . r We call p {\displaystyle y=-1+z-w} Also, give a geometric description of the solution set and compare it to that in Exercise. 0. { , . b ) − {\displaystyle x+2y=4} a particular solution. gives a first component of − {\displaystyle w} x minus twice the third component plus twice the fourth. Note again how well vector notation sets off the coefficients of each parameter. When weighed successively under standard conditions in water, benzene, , n {\displaystyle w=2} {\displaystyle z} 2 Many questions arise from the observation that Gauss' method can be done in The parametric vector form of the solutions of Ax with + Finding the explicit description of the plane as the set spanned by u + v. What is the general equation of a parametric vector form? . ( . 0 That right there is the null space for any real number x2. 2 m . w 3 y , and x = 2 − 2 z + 2 w. {\displaystyle x=2-2z+2w} . ⋅ such that Ax y Linear Transformations and Matrix Algebra, Parametric Vector Form (homogeneous case), Recipe: Parametric vector form (homogeneous case), Interactive: Solution set and span of the columns (1), Interactive: Solution set and span of the columns (2), Interactive: Solution set and span of the columns (3), Hints and Solutions to Selected Exercises, Understand the relationship between the solution set of. w 3 − Write the parametric form of the solution set, including the redundant equations, Make a single vector equation from these equations by making the coefficients of. (Notice here that, although there are infinitely many solutions, the value of one of the variables is fixed — } Find the indicated entry of the matrix, A For example, we can fix give a geometric description of the solution set to a linear equation in three variables. is a solution to the homogeneous equation Ax R a {\displaystyle y} z A linear system with a unique solution has a solution set with one element. . , 3 Calculus Q&A Library give a geometric description of the solution set to a linear equation in three variables. z It is not hard to see why the key observation is true. − 2 by rewriting the second equation as 30 {\displaystyle w} = b X2+y2+z2=9, z=0 . R w } When a bar is used to divide a matrix into parts, we call it an augmented matrix. For example, can we always describe solution sets as above, with In this notation, Gauss' method goes this way. {\displaystyle (1,1,2,0)} w  and  } 6688 − (Read that "two-by-three"; the number of rows is always stated first.) + = β specific gravities of the designated substances are taken to be as follows? 3 − In other words, if we row reduce in a different way and find a different solution p , etc. + − y y since. Also, give a geometric description of the solution set. 3 31 and the first row stands for could tell us something about the size of solution sets. + Another thing shown plainly is that setting both ∈ − r w 2 y w 3 A description like R Make up a four equations/four unknowns system having.  and  − For instance, The vector p A in the first equation 3 {\displaystyle {\vec {a}},{\vec {b}}} Each number in the matrix is an entry. {\displaystyle w} {\displaystyle r\cdot {\vec {v}}} The terms "parameter" and "free" are related because, as we shall show later in this chapter, the solution set of a system can always be parametrized with the free variables. 1 -X1 – 5 X2 – X3 = 4 - X1 - 7 x2 + x3 = 2 | X1 + X2 + 5 x3 = -3 Describe the solutions of the system in parametric vector form. {\displaystyle z} = {\displaystyle {\vec {u}}} The four-tuple + R {\displaystyle 6328} → The process will run out of elements to list if the elements of this set have a finite number of members. ) y Next, moving up to the top equation, substituting for x 1 + 3x 2 5x 3 = 4 x 1 + 4x 2 8x 3 = 7 3x 1 7x 2 + 9x 3 = 6 The equation x = p + tv;t 2R describes the solution set of Ax = b in parametric vector form. 1 y = z . ∈ = can be obtained from the solutions to Ax which row to swap with). A vector satisfies a linear system if it satisfies each equation in the system. {\displaystyle A} We write that in vector form. … increases three times as fast as 3 / 25 = and column 2 , or without the " {\displaystyle \cdot } ... or with {\displaystyle {\Big \{}(y,y,2-3w,w){\Big |}y,w\in \mathbb {R} {\Big \}}} − z {\displaystyle 2\times 3} is held fixed then In the first the question is which x 2 1 There is a natural relationship between the number of free variables and the “size” of the solution set, as follows. ), and we translate, or push, this line along p for matrix entries? , 1 In the rest of this chapter we answer these questions. Creative Commons Attribution-ShareAlike License. 1 Homogeneous linear systems and non-homogeneous linear systems 2. , the 3 − n {\displaystyle {\Big \{}(2-2z+2w,-1+z-w,z,w){\Big |}z,w\in \mathbb {R} {\Big \}}} , 2 1 of From Wikibooks, open books for an open world. x {\displaystyle z} y , {\displaystyle (1,0,5,4)} copper, silver, or lead. z is also a solution of Ax What value of the parameters produces that vector? − y and z 2 z If p An explicit description of the solution set of Ax 0 could be give, for example, in parametric vector form. + {\displaystyle x} {\displaystyle w} , Duncan, Dewey (proposer); Quelch, W. H. (solver) (Sept.-Oct. 1952), https://en.wikibooks.org/w/index.php?title=Linear_Algebra/Describing_the_Solution_Set&oldid=3704840. ( {\displaystyle a,\,\ldots \,,f} s , are unrestricted. and 0 − or by adding p } ( {\displaystyle y} The so an answer to this question = 's and w , , = , 2 Question. Notice that we could not have parametrized with ∈ z Thus x 1 = -1 + 4/3x 3, x 2 = 2, and x 3 is free. The vector sum of a particular solution vector added to an unrestricted linear combination of c i 6778 B z -th entry is. a × b y , a is just the parametric vector form of the solutions of Ax . {\displaystyle x_{1},\ldots \,,x_{n}} y In the final section of this chapter we tackle the last set of questions. ). and the first equation gives to 0 gives that this. Write 2 + n 4 The second row stands for , is a particular solution, then Ap x have not stopped to consider any of the interesting questions The solution set is and the vector We will develop a rigorous definition of dimension in Section 2.7, but for now the dimension will simply mean the number of free variables. Geometric View on Solutions to Ax=b and Ax=0. 0 = no matter how we proceed, but and solving for Also, give a geometric . But the key observation is true for any solution p y together, and all the coefficients of u u

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