Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN: 9781285463247
Author: David Poole
Publisher: Cengage Learning
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[画像:One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' x1,...,, since one actually computes only with the coefficients A,, and the scalars y.. We shall now abbreviate the system (1-1) by Do not solve using AI, I want real solutions with graphs and codes, wherever required. Reference is given, if you need further use hoffmann book of LA or maybe Friedberg. where AX = Y An Ain A = Aml Amn 21 31 X and Y = Problem 4: Proof and Application of Elementary Row Operations Given: Let D be a matrix of order n xn, where n ≥ 3, and assume the matrix can be written as: We call A the matrix of coefficients of the system. Strictly speaking, the rectangular array displayed above is not a matrix, but is a repre- sentation of a matrix. An m X n matrix over the field F is a function A from the set of pairs of integers (i, j), 1≤i≤m, 1≤ j ≤ n, into the field F. The entries of the matrix A are the scalars A(i,j) = A., and quite often it is most convenient to describe the matrix by displaying its entries in a rectangular array having m rows and n columns, as above. Thus X (above) is, or defines, an n XI matrix and Y is an m x 1 matrix. For the time being, AX = Y is nothing more than a shorthand notation for our system of linear equations. Later, when we have defined a multi- plication for matrices, it will mean that Y is the product of A and X. We wish now to consider operations on the rows of the matrix A which correspond to forming linear combinations of the equations in the system AX Y. We restrict our attention to three elementary row operations on an m x n matrix A over the field F: 1. multiplication of one row of A by a non-zero scalar e; 2. replacement of the rth row of A by row r plus c times row 8, c any scalar and rs; 3. interchange of two rows of A. Tasks: 011 012 021 022 1m 029 D= anl an2 ann 1. Prove that any square matrix can be transformed into an upper triangular form using a finite number of elementary row operations. Provide a step-by-step explanation and illustrate the proof with an example where n = 4. 2. Using your proof, explain why the determinant of a matrix can be computed from its upper triangular form. Demonstrate this process by calculating the determinant of a given 4 x 4 matrix using row reduction. 3. Prove that if a matrix D can be reduced to a matrix with a row of zeros, then the determinant of D is zero. Illustrate this with an example. 4. Show how the elementary row operations affect the eigenvalues of a matrix and provide a proof for why row operations do not generally preserve eigenvalues, except in specific cases. 5. Visualize the effect of elementary row operations on matrix D by considering the transformation of a unit square (or hypercube in higher dimensions) under these operations. Provide graphical representations for the transformations in 2D or 3D cases.]
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Transcribed Image Text:One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' x1,...,, since one actually computes only with the coefficients A,, and the scalars y.. We shall now abbreviate the system (1-1) by Do not solve using AI, I want real solutions with graphs and codes, wherever required. Reference is given, if you need further use hoffmann book of LA or maybe Friedberg. where AX = Y An Ain A = Aml Amn 21 31 X and Y = Problem 4: Proof and Application of Elementary Row Operations Given: Let D be a matrix of order n xn, where n ≥ 3, and assume the matrix can be written as: We call A the matrix of coefficients of the system. Strictly speaking, the rectangular array displayed above is not a matrix, but is a repre- sentation of a matrix. An m X n matrix over the field F is a function A from the set of pairs of integers (i, j), 1≤i≤m, 1≤ j ≤ n, into the field F. The entries of the matrix A are the scalars A(i,j) = A., and quite often it is most convenient to describe the matrix by displaying its entries in a rectangular array having m rows and n columns, as above. Thus X (above) is, or defines, an n XI matrix and Y is an m x 1 matrix. For the time being, AX = Y is nothing more than a shorthand notation for our system of linear equations. Later, when we have defined a multi- plication for matrices, it will mean that Y is the product of A and X. We wish now to consider operations on the rows of the matrix A which correspond to forming linear combinations of the equations in the system AX Y. We restrict our attention to three elementary row operations on an m x n matrix A over the field F: 1. multiplication of one row of A by a non-zero scalar e; 2. replacement of the rth row of A by row r plus c times row 8, c any scalar and rs; 3. interchange of two rows of A. Tasks: 011 012 021 022 1m 029 D= anl an2 ann 1. Prove that any square matrix can be transformed into an upper triangular form using a finite number of elementary row operations. Provide a step-by-step explanation and illustrate the proof with an example where n = 4. 2. Using your proof, explain why the determinant of a matrix can be computed from its upper triangular form. Demonstrate this process by calculating the determinant of a given 4 x 4 matrix using row reduction. 3. Prove that if a matrix D can be reduced to a matrix with a row of zeros, then the determinant of D is zero. Illustrate this with an example. 4. Show how the elementary row operations affect the eigenvalues of a matrix and provide a proof for why row operations do not generally preserve eigenvalues, except in specific cases. 5. Visualize the effect of elementary row operations on matrix D by considering the transformation of a unit square (or hypercube in higher dimensions) under these operations. Provide graphical representations for the transformations in 2D or 3D cases.
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