One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' 1,,, 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 Au Ain A = A 21 Y1 X and Y Statement: The rank-nullity theorem states that for any matrix A of size m x n, the sum of the rank and the nullity of A equals n: rank(A)+nullity(A) = n 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 X I matrix and Y is an m X I 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 I 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 r8; 3. interchange of two rows of A. Tasks: 1. Provide a general proof of the rank-nullity theorem using the concepts of linear transformations, dimension of vector spaces, and the properties of linear maps. Explain each step in detail and show how the theorem arises naturally from these concepts. 2. Discuss the implications of the rank-nullity theorem for the solutions of a linear system Ax = b . Explain how the rank of the matrix relates to the number of solutions and under what conditions the system has a unique solution, infinitely many solutions, or no solution. 3. Graphically illustrate the relationship between the column space, null space, and row space of a matrix. Use diagrams to show how the dimensions of these spaces relate to the rank and nullity. 4. Consider a matrix A where the null space is non-trivial (i.e., contains non-zero vectors). Explain why such a matrix cannot be of full rank and provide a theoretical argument to support this observation. Include a discussion on the geometric interpretation of the null space.
One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' 1,,, 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 Au Ain A = A 21 Y1 X and Y Statement: The rank-nullity theorem states that for any matrix A of size m x n, the sum of the rank and the nullity of A equals n: rank(A)+nullity(A) = n 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 X I matrix and Y is an m X I 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 I 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 r8; 3. interchange of two rows of A. Tasks: 1. Provide a general proof of the rank-nullity theorem using the concepts of linear transformations, dimension of vector spaces, and the properties of linear maps. Explain each step in detail and show how the theorem arises naturally from these concepts. 2. Discuss the implications of the rank-nullity theorem for the solutions of a linear system Ax = b . Explain how the rank of the matrix relates to the number of solutions and under what conditions the system has a unique solution, infinitely many solutions, or no solution. 3. Graphically illustrate the relationship between the column space, null space, and row space of a matrix. Use diagrams to show how the dimensions of these spaces relate to the rank and nullity. 4. Consider a matrix A where the null space is non-trivial (i.e., contains non-zero vectors). Explain why such a matrix cannot be of full rank and provide a theoretical argument to support this observation. Include a discussion on the geometric interpretation of the null space.
Chapter4: Systems Of Linear Equations
Section4.6: Solve Systems Of Equations Using Determinants
Problem 279E: Explain the steps for solving a system of equations using Cramer’s rule.
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Question
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'
1,,, 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
Au
Ain
A =
A
21
Y1
X
and Y
Statement: The rank-nullity theorem states that for any matrix A of size m x n, the sum of the rank
and the nullity of A equals n:
rank(A)+nullity(A) = n
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 X I matrix and Y is an m X I 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 I 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 r8;
3. interchange of two rows of A.
Tasks:
1. Provide a general proof of the rank-nullity theorem using the concepts of linear transformations,
dimension of vector spaces, and the properties of linear maps. Explain each step in detail and
show how the theorem arises naturally from these concepts.
2. Discuss the implications of the rank-nullity theorem for the solutions of a linear system Ax = b
. Explain how the rank of the matrix relates to the number of solutions and under what
conditions the system has a unique solution, infinitely many solutions, or no solution.
3. Graphically illustrate the relationship between the column space, null space, and row space of a
matrix. Use diagrams to show how the dimensions of these spaces relate to the rank and nullity.
4. Consider a matrix A where the null space is non-trivial (i.e., contains non-zero vectors). Explain
why such a matrix cannot be of full rank and provide a theoretical argument to support this
observation. Include a discussion on the geometric interpretation of the null space.
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