Write down the compatibility conditions on the following...

Write down the compatibility conditions on the following systems of linear equations by first computing a basis for the cokernel of the coefficient matrix. (a) $2 x+y=a$, $x+4 y=b,-3 x+2 y=c$; (b) $x+2 y+3 z=a,-x+5 y-2 z=b, 2 x-3 y+5 z=c$; (c) $x_1+2 x_2+3 x_3=b_1, x_2+2 x_3=b_2, 3 x_1+5 x_2+7 x_3=b_3,-2 x_1+x_2+4 x_3=b_4$; (d) $x-3 y+2 z+w=a, 4 x-2 y+2 z+3 w=b, 5 x-5 y+4 z+4 w=c, 2 x+4 y-2 z+w=d$.

Write down the compatibility conditions on the following systems of linear equations by first computing a basis for the cokernel of the coefficient matrix. (a) $2 x+y=a$, $x+4 y=b,-3 x+2 y=c$;
(b) $x+2 y+3 z=a,-x+5 y-2 z=b, 2 x-3 y+5 z=c$;
(c) $x_1+2 x_2+3 x_3=b_1, x_2+2 x_3=b_2, 3 x_1+5 x_2+7 x_3=b_3,-2 x_1+x_2+4 x_3=b_4$;
(d) $x-3 y+2 z+w=a, 4 x-2 y+2 z+3 w=b, 5 x-5 y+4 z+4 w=c, 2 x+4 y-2 z+w=d$.

Key Concepts

Compatibility Conditions

Compatibility conditions are restrictions on the right-hand side of a linear system that must be satisfied for the system to have a solution. These conditions are derived by asserting that the right-hand side vector must lie within the range or image of the coefficient matrix. The cokernel basis provides a concrete way to express these conditions, by requiring that the inner product between the right-hand side and each cokernel basis vector is zero.

Linear Systems of Equations

A linear system is a collection of one or more linear equations involving the same set of variables. The goal when dealing with such systems is to determine whether there exists an assignment of values to the variables that satisfies all the equations simultaneously. Understanding the structure of the system, especially in relation to its coefficient matrix, is fundamental in determining the existence and uniqueness of solutions.

Coefficient Matrix

The coefficient matrix is formed by the coefficients of the variables in a system of linear equations. This matrix encapsulates the linear transformation applied to the variable vector and plays a crucial role in analyzing the properties of the system, such as its rank, nullspace, and ultimately, the conditions under which the system is solvable.

Cokernel

The cokernel of a linear transformation, or equivalently of the associated coefficient matrix, is defined as the quotient of the codomain by the image of the transformation. In the context of linear systems, the cokernel represents the collection of vectors (or data) that are orthogonal to all solutions of the homogeneous adjoint system. Its structure provides insight into the compatibility of the system’s equations with a given right-hand side.

Basis of the Cokernel

Finding a basis for the cokernel involves determining a set of vectors that spans the entire cokernel and is linearly independent. This basis is essential because it allows us to express any element of the cokernel as a linear combination of these basis vectors. In the context of compatibility conditions, the basis vectors are used to derive the necessary and sufficient conditions on the right-hand side of the system for a solution to exist.

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