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Instant-use add-on functions for the Wolfram Language
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Determine the consistency equations required for a system of linear equations to have a solution
ResourceFunction["LinearConstraints"][mat,vec] determines the constraint equations that a vector vec must satisfy for the matrix equation mat.x==vec to have a solution. | |
Find the constraint equations that the vector {a,b,c} must satisfy to be in the column space of the matrix:
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When the second argument is a symbol, the second argument is replaced by a subscripted vector in that symbol:
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When Subscript→False, the second argument is replaced by the array vector {b[1],b[2],b[3]}:
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Setting the option PrintDisplay→True prints the matrix form of the reduced matrix that produced the constraint equations:
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The print statement generates a matrix displayed in MatrixForm. To convert it to a matrix, paste a copy of the matrix into a new cell and select Cell → ConvertTo → StandardForm (Shift + Ctrl + N):
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When there are no constraint equations (i.e. when the system is always consistent), the empty list is returned:
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The column space of a matrix is the null space of the coefficient matrix of its constraint equations. Begin with a matrix with rank 3:
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The constraint equations are:
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The coefficient matrix of the constraint equations is:
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The null space of this matrix is:
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These vectors are independent. To see that they are in the column space of the matrix A, simply verify that they satisfy the constraint equations:
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The left null space of a matrix is the row space of the coefficient matrix of its constraint equations. Begin with a matrix with rank 3:
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The constraint equations are:
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The coefficient matrix of its constraint equations is:
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The rows of this matrix span the left null space of A, since they are independent and satisfy BA=0:
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