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eaton rtlo 14713 service manualShed the societal and cultural narratives holding you back and let step-by-step Linear Algebra textbook solutions reorient your old paradigms. NOW is the time to make today the first day of the rest of your life. Unlock your Linear Algebra PDF (Profound Dynamic Fulfillment) today. YOU are the protagonist of your own life. Let Slader cultivate you that you are meant to be! Please reload the page. Shed the societal and cultural narratives holding you back and let step-by-step Linear Algebra and Its Applications textbook solutions reorient your old paradigms. NOW is the time to make today the first day of the rest of your life. Unlock your Linear Algebra and Its Applications PDF (Profound Dynamic Fulfillment) today. The Sponsored Listings displayed above are served automatically by a third party. Neither the service provider nor the domain owner maintain any relationship with the advertisers. In case of trademark issues please contact the domain owner directly (contact information can be found in whois). By clicking “Accept”, you consent to the use of ALL the cookies. Cookie settings ACCEPT Manage consent Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information. It is mandatory to procure user consent prior to running these cookies on your website. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate.http://happybomtv.com/data/board/20201010135039.xml
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THIS MANUAL DOES NOT CONTAIN SOLUTIONS TO ALL PROBLEMS INSTANT DOWNLOAD: Wifizim Follow Mathematics More from Wifizim Follow More From Medium Math for Data Science Beginners: Plotting and Data Visualization Benjamin Obi Tayo Ph.D. Commutative property Michele Diodati in Not Zero 6 Brain Damaging Habits You May Want to Quit Shivendra Misra in Publishous From The Ex-Wife to the New Wife: Your Old Position of Side Chick Has Been Filled Toni Crowe in ILLUMINATION This is the Future of Webdev. Crazy, Right? Ilya Suzdalnitski I tripled my income with data science. Here’s how. Natassha Selvaraj in Towards Data Science 20 high-paying, low-stress jobs that are growing Business Insider in Business Insider Four Fake Nice Gestures That Are Actually Manipulative Sean Kernan in Mind Cafe About Write Help Legal Get the Medium app. I can (F10) (Sp11), Introduction to Linear Algebra, (New 4th Edition), by Gilbert Strang. Chapter 1: Systems of Linear Equations and Matrices (another solution: interchange the first row and the third row to obtain. Publisher: University of Pennsylvania. Main Linear Algebra Solutions manual. Send-to-Kindle or Email. Linear Algebra Igor Yanovsky, 2005 2 Disclaimer: This handbook is intended to assist graduate students with qualifying examination preparation. Year: 2017. The answers in this manual supplement those given in the answer key of the textbook. PDF form without charge to students at schools adopting the 11th edition of the text. Introduction This book is an excellent introduction to linear algebra and is the consistent textbook for Massachusetts Institute of Technology's (MIT) linear algebra course 18.06.If you plan to seriously study any of the physical sciences: mathematics, physics,engineering etc. Edition: 2nd. File: PDF, 592 KB. david lay solutions manual linear algebra and its applications 4th edition manual pdf linear algebra with applications solution manual pdf gilbert strang.ISBN 10: 981-02-0395-0.http://www.spuni.cz/files/f430-manual-or-auto.xml Solutions Manual Applied Linear Algebra 2nd edition by Peter J. Olver, Chehrzad Shakiban Read Solution Manual Linear Algebra PDF on our digital library. You can read Solution Manual Linear Algebra PDF direct on your mobile phones or PC. As per our directory, this eBook is listed as SMLAPDF-106, actually introduced on 30 Jan, 2021 and then take about 1,579 KB data size.By rows or by columns: 9 separate multiplicationswhen A is 3 by 3. Student Solutions Manual for Calculus for the Life Sciences - (OPTIONAL) by R. Greenwell, Linear Algebra and its Applications, 3rd Edition, by David Lay. Page Count: 485 University of MarylandThese efforts include theThe author andThe author and publisher shall not be liable in any event for incidental or consequentialReproduced by Pearson Addison-Wesley from electronic files supplied by the author. Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in anyPrinted in the United States of America.Additional notes are at the end of the section.The next two steps should be to use the variable x3 in the third equation toIn matrix notation, that means to replace R2 byFurther row. The solution set is empty. Publishing as Addison-Wesley.Use the 1 in the fourth row to change the 3Publishing as Addison-Wesley.In fact, using the argument from Example 2,Publishing as Addison-Wesley.The solution set consists of all possiblePublishing as Addison-Wesley.So the original system has a solutionOtherwise, for some choices of fHere are threeThe others are:Publishing as Addison-Wesley.As a result,Exercise 14 in Section 1.10 returns to this problem and states the solution in case students have notExercise 31 in Section 2.5 uses this same type of problem inFor instructors who wish to use technology in the course, the Study Guide provides boxed MATLABThe MATLAB box for SectionThis featureThese commands are included in the text data sets, available from the text web site. Publishing as Addison-Wesley.The remaining variable x2 isThe general solution isPublishing as Addison-Wesley.Corresponding system of equations: 1The remaining variable x3 isIn this particular problem, the basicTo avoid this, identify the basicCorresponding system:Corresponding system:Corresponding system:There are many solutions because x3 is a free variable.There are many solutions because x1 is a free variable.Publishing as Addison-Wesley.There are many solutions because x2 is a free variable.Thus the system has a solution for all values of h.There are no free variables.There are no free variables.False. See Theorem 1. False. See the second paragraph of the section. True. Basic variables are defined after equation (4). True. This statement is at the beginning of Parametric Descriptions of Solution Sets.The pivot positions in a matrix areSee the solution of Practice. Problem 2. Publishing as Addison-Wesley.No matter what the values of a, b, c, and d, the solution exists and is unique.Since the matrix is the augmented matrix for a system, Theorem 2 showsSo, the system is consistent, by. Theorem 2.This statement is true because the free variables correspond to nonpivot columns of the coefficientAnd there are noThere cannot be more basicSuch a variable may beIf the system is consistent, each different value of a freeThe fraction associated with the backward phase is about. Publishing as Addison-Wesley.Row reduce theExercise 34 requires 25 row operations. It should give students an appreciation for higher-levelThe commandThe Study Guide includes a “Mathematical Note” about the phrase, “If and only if,” used in TheoremUsing the definitions carefully,Usually the intermediate steps are not displayed.Since the grid can be extended in every direction, the figure suggests that everyTo write a vector a as a linear combination of u and v, imagine walking from the origin to a alongPublishing as Addison-Wesley.Another solution isFigure for Exercises 7 and 8Since the grid can be extended in every direction, the figure suggests that everyIf you prefer to stay on the paths displayed on the “map,”Usually, the intermediate calculations are not displayed. Publishing as Addison-Wesley.Is b a linear combination of a1, a2, and a3?The equationPublishing as Addison-Wesley.To determine if b is a linear combination of these columns,Row reduce the augmented matrixNote: Exercises 19 and 20 prepare the way for ideas in Sections 1.4 and 1.7.Publishing as Addison-Wesley.True. See the beginning of the subsection Vectors in Rn. False. Setting all the weights equal to zero results in a legitimate linear combination of a set ofTo determine if b is in W, use thePublishing as Addison-Wesley.By definition,Publishing as Addison-Wesley.Here c1, c2,The darker parallelogram shows that b is a linear combination of v1The larger parallelogram shows that b is a linear combination of v1 and v3, that is,By associativity of addition in R, this. By definition of equality ofThese entries are equal, by a distributiveNote: When an exercise in this section involves a vector equation, the corresponding technology data (inTo use MATLAB or otherThe MATLAB note inThe appendices in the Study Guide give correspondingPublishing as Addison-Wesley.Exercise 34Write the matrix A whosePublishing as Addison-Wesley.Finding the solution by handNote: The skill of writing a vector equation as a matrix equation will be important for both theory andSee also Exercises 27 and 28.Write the matrix A whosePublishing as Addison-Wesley.The set of b forSo all four statements in. Theorem 4 are false. Thus, not all vectors in R4 can be written as a linear combination of the columnsPublishing as Addison-Wesley.So all four statements inFalse. See the paragraph following equation (3). True. See the box before Example 3. False. See the warning following Theorem 4. True. See Example 4. True. See parts (c) and (a) in Theorem 4. True. In Theorem 4, statement (a) is false if and only if statement (d) is also false.Publishing as Addison-Wesley.Perform a row operationThe justification is given by row reducing A to B, in order to display thePerform a row operation that creates a matrix A that is not in echelon form. Then A has the desiredTheorem 4.With only two columns, A can have at most two pivotGenerally, if A is an m?n matrixReason: the matrix A whose columns are these threeTo have a pivot in each row, A would have to have at least four columns (oneSince A does not have a pivot in every row, its columns do notHowever, these exercises serve to review ideas from Section 1.2,Publishing as Addison-Wesley.The original matrix doesTheorem 4.Notice that the fourth column of the original matrix,Let Ao be the matrix formed by deleting column 4 of A, let B be theSince Bo is in echelon form, itTherefore, the columns of Ao span R4. It is possible to delete column 3 of A instead of column 4. In this case, the fourth column of ANote: Exercises 41 and 42 help to prepare for later work on the column space of a matrix. (See SectionIf a column of A is removed, forming a newThe third column of the original matrix, say A, is not aSince Bo is in echelon form, it shows that Ao hasTherefore, the columns of Ao span R4. It is possible to delete column 2 of A instead of column 3. (See the remark for Exercise 41.). However, only one column can be deleted. If two or more columns were deleted from A, theNotes: At the end of Section 1.4, the Study Guide gives students a method for learning and masteringSpecific directions are given for constructing a review sheet that connects theI require my students to prepare such a sheet thatLater, the students use these sheets when studying for exams. The MATLAB box for Section 1.4 introduces two useful commands gauss and bgauss thatThe commandThe result is aI use the gauss command frequently in lecturesPublishing as Addison-Wesley.This command,The use of gauss and bgauss creates an environment in which students use their computerUnless you are able to conduct yourFigure 6 will appearFor solving homogeneous systems, the text recommends working with the augmented matrix, although no calculations take place in the augmented column. See the Study Guide comments on Exercise 7If you do notExercises 35 and 36 help. Exercise 37 will help students avoid the standard mistake ofIf a column of theOtherwise, the system has only the trivial solution.With only two equations,One variable must be free. Refer to Exercise 29 in SectionPublishing as Addison-Wesley.The remaining variables are free. In particular, x4 is free (andIn parametric vector form. Publishing as Addison-Wesley.Note: The Study Guide discusses two mistakes that students often make on this type of problem.The general solution isThe solution set is the line through p in the direction of q.In vector form,The solution set of the homogeneous equation is the plane through the origin in R3 spanned byIn vector form,Publishing as Addison-Wesley.See the subsectionThe box before Example 1 uses theThe statement (from Theorem 6) is true only when thereThere are two things to prove: (a) every vector. By Theorem 5(a) in section 1.4Publishing as Addison-Wesley.By Theorem 4 inBy Theorem 4So the equationSo the equation AxPublishing as Addison-Wesley.For b, takeNote: In the Study Guide, a “Checkpoint” for Section 1.5 will help students with Exercise 37. Now, let c and d be scalars. Using both parts of Theorem 5. Since A is 3?3, it has at most. Note: The MATLAB box in the Study Guide introduces the zeros command, in order to augment aThe entries in a column describe where a sector'sDistribution of. Output From. Goods Services. Purchased by:Denote the total annual output (in dollars) of the sectors by pG and pS. From the first row, the total. So the equilibriumPublishing as Addison-Wesley.Move all variables to the left side and combine like terms. Row reduce the augmented matrix:Only the ratio ofThe other equilibrium prices are then. Any constant nonnegative multiple of these prices is a set ofThe ratios of the prices remain the same,The entries in a column describe where a sector’sDistribution of Output From:Fuels and Power. Manufacturing. ServicesFuels and Power. ServicesFrom the first row ofManufacturing sector and the Services sector are, respectively. Publishing as Addison-Wesley.If pS is assigned the value of 100,The economic equilibrium is not affected by a proportional change inThe entries in each column must sum to 1. Distribution of Output From:Mining. Lumber. Energy. TransportationMining. Transportation. From the firstFrom the second, third, and fourth rows of the table, the equilibrium equations are. Move all variables to the left side and combine like terms. Reduce the augmented matrix to reduced echelon form. Publishing as Addison-Wesley.The entries in each column must sum to 1. Distribution of Output From. Purchased by. Agriculture. Services. TransportationTransportationAgricultureMove all variables to the left side and combine like terms. Reduce the augmented matrix to reduced echelon form:The entries in each column must sum to 1. Publishing as Addison-Wesley.Purchased by. Reduce the augmented matrix to reduced echelon form:Manufacturing sectirs to go down slightly, while increasing the unit price for the Services sectorPublishing as Addison-Wesley.The order of the various atoms is not important. The list here was selected by writing the elements inThe balanced equation is. Using the order manganese (Mn), sulfur (S),The balanced equation isEverything balances except the hydrogen. The right side is short 1 hydrogen atom. Perhaps theIntersection. Flow inFlow out. Publishing as Addison-Wesley.Reduce the augmented matrix:Since x3 cannot be negative, the minimum value of x4 is 80.Intersection Flow in. Flow outThe minimum flows areIntersectionRearrange the equations:IntersectionRearrange the equations:Flow outNote: The MATLAB box in the Study Guide discusses rational calculations, needed forAs usual, the appendices cover thisPublishing as Addison-Wesley.There is a danger, however, that students will memorize the result without understanding the proof, andExercises 37 and 38 anticipate the discussion in Section 1.9Exercise 44 is fairly difficult for my students.By comparing entries of the. Thus, the two vectors are linearlyBut there is a sign problem with the second entries. So neither of the vectors isThe vectors are linearly independent.There are only 3 rows, so there are at most three pivot positions. Hence, atThus, the vectors are linearly dependent if and only. Publishing as Addison-Wesley.But this multipleFalse. A homogeneous system always has the trivial solution. See the box before Example 2. False. See the warning after Theorem 7. True. See Fig. 3, after Theorem 8. True. See the remark following Example 4.Publishing as Addison-Wesley.B: any 3?2 matrix with two nonzero columns such that neither column is a multiple of the other. InTake v3 to be not a multiple ofThis problem canThen the trivial solution is the only solution, and so thePublishing as Addison-Wesley.Since B itself was constructed to have four pivotNote: At the end of Section 1.7, the Study Guide has another note to students about “Mastering Linear. Algebra Concepts.” The note describes how to organize a review sheet that will help students form aThe note also lists typical misuses of terminology, in which anPublishing as Addison-Wesley.Hopefully, theseExercise 25 (and 26) can be used to make a few comments about computerThen the Guide explains how to createExercises 19 and 20 provide a natural segue into Section 1.9. I arrange to discuss the homework onThe text does not provide much practice determining whether a transformation is linear, because theIf you want yourIf you skip the concepts of one-to-one and “onto” in Section 1.9, you can use the result of Exercise 31The value of b is 6. The codomain of T is R6, because Ax is a linearFor the domain of T to be R5, A must have fiveFor the codomain of T to be R7, the columns of A mustA scaling by the factor 2. The transformation in Exercise 13 may also be described as a rotation of.A scaling by a factor of 2 and a projection onto the x2Locate T(u)T(w). T(v)T(u). Publishing as Addison-Wesley.The key idea is to writeSee the remark about Example 1(d), following theIn this case, for all x,y in R and all scalars c and d. This shows that f is linear.If b is nonzero, then f (2x) is notPublishing as Addison-Wesley.To summarize,Since a linearPublishing as Addison-Wesley.Note that since T(x) and x have four entries, A must be aSince x has 2 entries, A has 2 columns. Since. T(x) has 4 entries, A has 4 rows.Since x has 3 entries, A has 3 columns.Since x has 4 entries, A has 4 columns.Publishing as Addison-Wesley.True. See Theorem 10. True. See Example 3. False. See the paragraph before Table 1. False. See the definition of onto. Any function from Rn to Rm maps each vector onto anotherFalse. See Theorem 12. True. See Theorem 10. False. See the definition of one-to-one. Any function from Rn to Rm maps a vector onto a singleAlso, since A does not have a pivot in each row, the columns of A do not span R4. By Theorem 12, TSo T is not one-to-one, by Theorem 12. Also, A isPublishing as Addison-Wesley.So T is not one-to-one, by TheoremBy Theorem 12, T maps R3 onto R2.So T is one-to-one, byAn alternate argument for the second part is to observe directly fromThis is more or less evident, based on experienceSo each column of A must be a pivot column:There are four possibilities for the echelon form. Note that T cannot be one-to-one because of the shape of A.And from the statement in Exercise 30 in Section 1.7,Thus, “T maps Rn onto Rm if and only A hasHowever, by matrix-vectorSee equation (4) in Section 1.8. Publishing as Addison-Wesley.That is, m n.By Theorem 12, the transformation T does notBy Theorem 12, the transformation T maps R5 onto R5.Cereal, then x1 and x2 should satisfyPublishing as Addison-Wesley.Natural Cereal.Nutrients:Publishing as Addison-Wesley.The values of x1, x2, and x3 should satisfyFrom the given data,To solve, row reduce the corresponding augmented matrix:Publishing as Addison-Wesley.Nutrient. Nutrients. RequiredThese amounts must satisfy the following matrix equationAlthough theThe mixture of.64Voltage drop for I 2 is negative; I 2 flows in opposite direction. Current I 3 does not flow in loop 1. Current I4 does not flow in loop 1. Loop 2: The resistance vector isTotal of RI voltage drops for current I 2. Voltage drop for I 3 is negative; I 3 flows in opposite direction. Current I 4 does not flow in loop 2This happens because the loop currentVoltage drop for I 2 is negative; I 2 flows in opposite direction. Current I 4 does not flow in loop 1. Current I 4 does not flow in loop 2Voltage drop for I 2 is negative; I 2 flows in opposite direction. Voltage drop for I 4 is negative; I 4 flows in opposite direction. Voltage drop for I 5 is negative; I 5 flows in opposite direction. Loop 2: The resistance vector is. Publishing as Addison-Wesley.Total of RI voltage drops for current I 2. Current I4 does not flow in loop 2. Voltage drop for I5 is negative; I5 flows in opposite directionThe statement that “about 7 of the city’s population moves to theThis determinesFrom. City Suburbs. To:Suburbs. Likewise, if 5 of the suburban population moves to the city, then the other 95 remain in theFrom. City Suburbs To:Suburbs. The remaining entries are determined by the fact that the numbers in each column must sum to 1.It is reasonable, but not essential, to consider the people livingThat is, the first entry in a column or row of a vector will concern the peopleWith this choice, the migration matrix has the form. From. Calif. Outside To:OutsideFor the second column of M, computeCalif. Outside To:OutsidePublishing as Addison-Wesley.Here are the vectors x0 through x6 with the first 5 figuresIn this case, the data inPublishing as Addison-Wesley. For Fig. (a), the equations are. To solve the system, rearrange the equations and row reduce the augmented matrix. InterchangingThe first five steps are shown in detail.Rearrange the equations and row reduce the augmented matrix:Fig. (a) in Exercise 14 of Section 1.10:Fig. (a):A verification of this statement is not expected. However,The system of equations that approximate the interior temperatures can. Publishing as Addison-Wesley.Note: The MATLAB box in the Study Guide for Section 1.10 discusses scientific notation and shows. Chapter 1 SUPPLEMENTARY EXERCISESSome counterexample is needed.)If A has m pivot positions, then A has mLet A be any matrix with m pivot columns but moreThus the equation does not does not have a unique solution. Publishing as Addison-Wesley.The reduced echelon form has a 1 in each pivot position, so the reduced echelon form is the n?nSection 1.7.See Theorem 7 inThis is impossible because A has only fiveSince A has n columns and m pivots, m might be less than n.Publishing as Addison-Wesley.Finally, if h v 12, theOtherwise, theEvery point in R2 can be described uniquely in terms of this grid. Thus, bIf the coefficient matrix is 2?3, thenMake one row replacement operation on the secondIf the coefficient matrix is 2?3, then onlyThe echelon form will have all zeros in the second row. Use aObserve that v1 is not the zero vector. Next, v2 is not a multiple of v1 because the third entry of v2The “first” vector v1 is nonzero, v2 is not aFinally, by looking at first entries in the vectors, v4By Theorem 7 in Section 1.7, the columns areThe first is a “direct” proof. The second is called a “proof by contradiction.”By hypothesis, v4 is not a linearThen by Theorem 7, one of the vectors in theThis vector cannot be v4 because v4 is not inPublishing as Addison-Wesley. Since v3 is on the line. So v3 is a linear combination of v1By Theorem 4 in Section 1.4, A has a pivotSince A has three columns, each column must be a pivot column. So the. By Theorem 12Publishing as Addison-Wesley.However, the only feasible solutions must have whole numbers of floors for each plan. Thus, x3B, and 8 floors of plan C. These are the only feasible solutions. A larger positive multiple of 8 forA negative value for x3, of course, is not feasible either. Publishing as Addison-Wesley.Exercises 23 and 24 are used in the proof of the Invertible Matrix Theorem, in Section 2.3. ExercisesOr, these exercises could be assigned after starting Secction 2.2. Exercises 27 and 28 are optional, buut they are mentioned in Example 4 of Section 2.4. OuPublishing as Addison-Wesley.Otherwise, AB is undefined. Since AB has 7That is, the first column of UQ lists the total costs for materials, labor, andColumns 2, 3,andPublishing as Addison-Wesley.See the boxThis is a common mistake.The order cannot be changed in general.By definition, the third column of AB is Ab3. ByThey are equal since b1 and b2 are equal.However, bp is not the zero vector, because B has no column of zeros.Note: The text answer for Exercise 21 is, “The columns of A are linearly dependent. Why?” The Study. Guide supplies the argument above, in case a student needs help. Since x is nonzero, the columns of AB mustSo every variable is a basic variable and every column of A is a pivot column.Publishing as Addison-Wesley.Note: Exercise 23 is good for mathematics and computer science students. The solution of Exercise 23 inThe Study Guide also has anThe details areNew statements will be addedPublishing as Addison-Wesley.A counterexample to the subspaceNote: Most students prefer to give a geometric counterexample, but some may choose an algebraic calculation. The four exercises here should help students develop an understanding of subspaces, but they mayDeveloping thatFor example, the sum of the vectors in H shownSee the diagram.Publishing as Addison-Wesley.Using A and p as in Exercise 7,All that is required here is one nonzero vector. So choose anyNote: Section 2.8 of Study Guide introduces the ref command (or rref, depending on the technology), which produces the reduced echelon form of a matrix. This will greatly speed up homework forAll that is required here is one nonzero vector. So choose any nonzeroThen A is invertible because itsPublishing as Addison-Wesley.Or, point out that theNote: The Study Guide warns students not to say that the two vectors here are a basis for R2.The critical phrases “for each” areThe condition about the zero vector isPublishing as Addison-Wesley.A wrong choice is to select columns 1 and 2 ofFor simplicity, the answers here and in the text list the vectorsThis style is also easier for students. I am careful,A shortcut, which some instructors might introduce later in theNamely, setBasis for Nul A:Note: The solution above illustrates how students could write a solution on an exam, when time isFor instance, if the first two columns of A are equal, then b could be (1,Since F is square, the IMT shows that F is not. That is, Nul F contains a nonzero vector. Since B is square, the IMT shows that B is invertible. Also, each solution is unique, by Theorem 5Since C is square, the IMT. Also, each solutionThis means that A cannot have moreAs a result, A must be a square matrix.So Col B is aBy Theorem 13, the pivotPublishing as Addison-Wesley.Publishing as Addison-Wesley.Coordinate vectors are important mainly to give an intuitive and geometric feeling for theIf you plan to omit Sections 5.4, 5.6, 5.7 and 7.2,The Rank Theorem in this section omits the nontrivial fact about Row A which is included in the. Rank Theorem of Section 4.6, but that is used only in Section 7.4. The row space itself can be introducedMy students need the practiceIn either case,Figure 1. Figure 2. Note: Figures 1 and 2 display what Secttion 4.4 calls B-graph paper. If so, thenFor Nul A, use the reduced echelon form, augmented with a zero column toR2, because Nul A is a subspace of R6. It is true, however, that Nul A is two-dimensional. Reason: theA line must be through the origin in Rn to be aIn this case, the spanning set is automatically a linearly independentSee Example 2.In this case, the linearly independent set is automatically a spanningPublishing as Addison-Wesley.Since a 5?7 matrix has seven columns, AThe remaining threeThe resulting matrix willThere is no need to worry about whether Nul A hasEvery column is a multiple of some fixedChoose any multiples of the vector for the other tthree columns.In particular, the columns are linearlyBy the Basis Theorem, this set ofNote that cj is in Rp because B has p columns. By hypothesis, q is larger than p, so C has more columns than rows. By a theorem, the columns ofPublishing as Addison-Wesley.Theorem that have been given so far. The format is the same as that used in Section 2.3, with threeFour statements are included that are not in the text’s official list ofThe Study Guide section also contains directions for making a review sheet for “dimension” andPublishing as Addison-Wesley.A has as many columns as B has rows.Matrix Theorem, A is invertible.Then ATA is the product ofBy Theorem 6 in. Section 2.2,By a property of matrix multiplication, shown after Example 6 in Section 2.1, and the fact that cThis shows that the columns of V are linearlySo, for every. L?col1(U) is a linear combination of the columns of L in which all weights except possibly the firstThus col2(A) is aNote: (In the Study Guide) It is possible that BA is invertible.