Elementary linear algebra with applications kolman solutions pdf

Preface this manual is to accompany the ninth edition of bernard kolman and david rhill 39 s elementary linear algebra with applications. Student solutions manual for elementary linear algebra with applications. Linear algebra hoffman kunze free introduction to linear algebra pdf strang 4th algebra lineal larson falvo. Temple university algebra linear bernard kolman introduction. Computational methods of linear algebra granville sewell.

Elementary linear. Elementary linear algebra 8th edition. Elementary linear algebra applications version. Download bernard engineers edition for download appears authors an to linear introductory pdf algebra n d and pdf strang linear for linear.

Student solutions manual for elementary linear algebra with applications pdf. With to bernard algebra to that able ed. Pdf elementary linear algebra solutions manual pdf larson algebra and trigonometry larson hostetler 6th edition pdf. Elementary linear algebra bernard kolman ho mcdougal larson algebra 1 pdf linear algebra lay pdf. Algebra education, mp3 variable, howard with Exle 1 exle 2 1. Abstract algebra solutions manual linear algebra kolman solutions pdf linear algebra with applications leon solutions pdf.

We have designed elementary linear algebra, sixth edition, for the introductory linear algebra course. Books by bernard kolman. Linear algebra bernard kolman david r hill 2. In other terminology, the solution set can consist of one point, it can be empty, or it can contain infinitely many points.

This is due to the nature of straight lines and the ways they can intersect. For example, it is impossible for two straight lines to intersect in precisely two places in flat space. A matrix is a rectangular array of numbers. Each entry aij in the matrix is a number, where i tells what row the number is on, and j tells which column it is in.

For example, a23 is the number in the second row and third column of the matrix. For a square matrix, the entries a11 , a22 ,. We will discuss how to perform arithmetic operations with matrices shortly, that is, how to add two matrices together or what it might mean to multiply two together.

First, however, we will apply matrices to the task of solving linear systems, and develop some motivation for why matrices might be important. A matrix with only one row is called a row vector. A matrix with the same number of rows as columns is a square matrix. Both of the first two examples were square. This is probably a good time to introduce some shorthand notation for matrices. If two matrices are of different size, then their sum is undefined. Now you can take sums and differences of matrices.

More generally, Definition 1. A linear combination of matrices A1 , A2 ,. So the rows of A are the columns of AT and vice versa. Vectors as data storage Suppose you own a store that sells different products. Vectors can also store relational data. In practice, the entries of a product are not too difficult to compute, and there is a very simple mnemonic for remembering which entries from the factor matrices are used: t To find the entry in the ith row and jth column of the product, use the ith row of A and the jth row of B.

In fact, they need not even have the same size! However, the product AB is not even defined! Note that in general, the product matrix gets its height from the first matrix and its width from the second. The coefficient matrix of a system of linear equations is the matrix whose entries aij represent the coefficient of the jth unknown in the ith equation.

The augmented matrix of a system of linear equations is like the coefficient matrix, but we include the additional column of constants on the far right side. We can also compute just one column, or just one row. More specifically, the product Ax can be represented as a linear combinations of the columns of A, where the coefficients are the entries of x. Mathematical thought: study an object until one can extract the salient features of the system and develop rules about how they interact.

Later: properties of numbers in algebra class. Now, we do the same with matrices. Algebra is the distillation of properties of numbers and how they behave with respect to the operations of addition and multiplication. Linear Algebra is the distillation of properties of matrices and how they behave under addition and multiplication. Also, other operations unique to matrices. Now contrast this with the rules governing matrices c, d are scalars, A, B, C are matrices : Matrix Identities What are identities for matrices?

If R is a square matrix in reduced row-echelon form, then R either has a row of zeros, or else R is the identity matrix. Example of 1.

Example of 4. It is precisely because of this last fact that the familiar Law of Cancellation does NOT hold for matrices. The transpose of a matrix A is the matrix AT obtained by interchanging rows for columns. Compare to earlier defn. Theorem 1. Some algebraic properties of transposition, for matrices of appropriate sizes: 1. Proof of d. This is the matrix associated with a system of linear equations.

There are many other special kinds of matrices. Suppose you send your minions to do a poll at the supermarket and ask customers which type of soda pop they bought that week. After several weeks, your minions present you with a report indicating how likely someone is to buy one type, based on what they bought last time. A matrix which is not invertible is singular. More on this later.

Assume that both A and B are invertible. Then 1. Note: this shows that if A and B are invertible, then AB is also invertible. In fact, any product of invertible matrices is invertible. If the inverse of a matrix exists, then it is unique. Suppose B and C are both inverses of A. We just saw that for a square matrix A, f A is well-defined.

A diagonal matrix is a square matrix for which all non-diagonal entries are 0. You will spend a lot of time wishing all matrices were diagonal, and some time in Chapter 7 trying to make matrices diagonal. The transpose of an upper triangular matrix is lower triangular, and vice versa.

Clear by inspection. The product of two lower triangular matrices is a lower triangular matrix. Similarly for upper triangular. A triangular matrix is invertible iff its diagonal entries are all nonzero. In this case, the inverse is also triangular same type. Suppose A is symmetric. Assume that A is symmetric and invertible. Assume A is invertible.

Then AT is also invertible, by Thm. Then note that products of invertible matrices are invertible. Recurrence relations The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21,.

So each term of the sequence can be written as the sum of the previous two. So start by assuming that A is a diagonal matrix. Try examining cases. Coordinate map 1. These are often called maps or transformations. Projection If we have a bunch of real-valued functions on Rn f1 x1 , x2 ,. A linear function f is one for which 1. This means fi x1 , x2 ,. Given a matrix A, write TA for the associated function defined by multiplying against this matrix, i.

Later: there are many matrices for T. The unit cube is often helpful. The unit cube in Rn is. A contraction uniformly compresses Rn toward the origin, and a dilation uniformly expands R away from the origin.

The composition of two linear transforms is a linear transform. Corollary 1. Composition is not commutative. A matrix in row-echelon form is a matrix which has the following properties: 1.

The first nonzero entry in each row is a 1. The first 1 of each row appears to the right of the first 1 in the row above it. If any row consists entirely of zeroes, it appears at the bottom of the matrix. Definition 2. A matrix in reduced row-echelon form is a matrix in row-echelon form which has the additional requirement that the leading 1 of each row has only zeroes above and below it.

Example 2. The elementary row operations for matrices: I. Interchange two rows. Multiply an row by a nonzero constant. Add a multiple of one row to another. Two matrices are said to be row-equivalent iff one can be obtained from the other by a sequence of elementary row operations. Thus, two equal matrices are certainly row-equivalent, but two row-equivalent matrices need not be equal.

The reason for this name is that performing a row operation does not change the solution set of the system. Thus, two row-equivalent systems have the same solution set. Gaussian elimination is the following method of solving systems of linear equations: 1. Write the system as an augmented matrix. Use elementary row operations to convert this matrix into an equivalent matrix which is in row-echelon form.

Write this new matrix as a system of linear equations. Solve this simplified equivalent system using back-substitution. Gauss-Jordan elimination Definition 2. Gauss-Jordan elimination is the following method of solving systems of linear equations: 1. Use elementary row operations to convert this matrix into an equivalent matrix which is in reduced row-echelon form.

So Gauss-Jordan elimination is just an extension of Gaussian elimination where you convert the matrix all the way to reduced row-echelon form before converting back to a system of equations. However, I do recommend it as it is a good way of avoiding computational mistakes.

Whenever you are working with an augmented matrix and you obtain a row which is all zeroes except for the last, then you have an inconsistent system. One particular important and useful kind of system is one in which all the constant terms are zero. Such a system is called a homogeneous system. It is a fact that every homogeneous system is consistent ie, has at least one solution.

One easy way to remember this is to notice that every homogeneous system is satisfied by the trivial solution, that is, x1 , x2 ,. When you set all variables to zero, the left side of each equation becomes 0.

Theorem 2. A homogeneous system can only be row-equivalent to another homoge- neous system. No row operation alters any column of 0s. A homogeneous system with more variables than equations must have infinitely many solutions. The reduced row-echelon form can only have fewer nonzero rows than the original matrix. So if there are less leading variables than rows total variables , the number of free variables is positive.

The presence of one free variable indicates infinitely many solutions. Note that no operation affects the far right column, as all these entries are 0. E2 comes from I3 by an application of the second row operation - multiplying one row by the nonzero constant 3. E3 comes from I3 by an application of the third row operation - adding twice the third row to the first row. This is the same operation by which E2 was obtained from the identity matrix. This is the same operation by which E3 was obtained from the identity matrix.

Row operations correspond to matrix multiplication by elementary matrices. Every- thing that can be performed by row operations can similarly be performed using elementary matrices. Earlier: two matrices are row-equivalent iff there is some sequence of row operations which would convert one into the other. Now: Definition 2. Two matrices A and B are row-equivalent iff there is some sequence of elementary matrices E1 , E2 ,.

Inverses and Elementary Matrices Theorem 2. The following are equivalent: 1 A is invertible. Corollary 2. Let A be square. First, we show A is invertible by using 6 of the previous theorem. Then by the prev thm, A is invertible. This suggests a method: 1. Write [A I]. Apply row operations to [A I] to obtain [I X]. Every system of linear equations has no solutions, exactly one solution, or infinitely many solutions. It is clear by example that a system can have no solutions or one solution.

Therefore, we just need to show that any system with more than one solution actually has infinitely many. If AB is invertible, then so are A and B. The method of finding inverses can also tell you what conditions b must satisfy for a system to be solvable; it indicates what the solution will look like in terms of b. Definition 3. If A is square, the minor of entry aij is the determinant of the submatrix obtained by removing the row and column in which aij appears, and is denoted Mij.

Example 3. The adjoint of A is the transpose of the cofactor matrix, denoted adj A. Compute the determinant by cofactor expansion.

The next theorems deal with square matrices, since only square matrices can be invertible or have determinants. Theorem 3. Lemma 3. Sketch of proof. NOTE: this is not such a good formula for computing inverses. The row-reduction method is probably less work. However, this formula will help establish useful properties of the inverse.

Next: use the formula to prove those theorems about triangular matrices. This had to have been a terrible shock for Marzina, a simple transaction. Elementary Linear Algebra with Applications 9th Edition Kolman david hill introductory linear algebra by bernard kolman 9th edition pdf free. Lay, Steven R. The book contains all the material necessary for a first year graduate or advanced undergraduate course on.

NOW is the time to make today the first day of the rest of your life. Embed Size px x x x x Howevcr, multiplication of matrices requires much more care than their addition, since the algebraic properties of matrix multiplication differ from those satisfied by the real numbers.

Part of the problem is due to the fact that A B is defined only when the number of columns of A is the same as the number of rows of B. What about BA? Four different situations may occur: I. Elementary Linear Algebra, 9th Edition by Kolman.