Relationships Between Planes
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1 Relationships Between Planes Definition: consistent (system of equations) A system of equations is consistent if there exists one (or more than one) solution that satisfies the system. System 1: {, System 2: {, and System 3: { are all consistent systems. They have corresponding solutions: System 1: ( ) ( ) System 2: ( ) ( ) System 3: all points satisfying the linear relation Definition: inconsistent (system of equations) A system of equations is inconsistent if there are no solutions that satisfy the system. System 1: {, System 2: {, System 3: { For each of these systems there does not exist a solution that satisfies each equation in the system. Definition: matrix A matrix is a two-dimensional (rectangular) array of numbers, symbols or expressions arranged in rows and columns. [ ], [ ], [ ] and [ ] are all matrices. Definition: size (of a matrix) The size of a matrix is based on the numbers of columns and rows in it. If a matrix has rows and columns then the size of the matrix if. [ ] is a matrix. Definition: equivalent (system of equations) Two systems of equations are equivalent if every solution to one system is also a solution to the second system, and vice versa. {, {, { and { are all equivalent systems.
2 Definition: coefficient matrix A coefficient matrix is a matrix which contains the coefficients of the variables of system of linear equations. Given the system {, the corresponding coefficient matrix is [ ]. Definition: augmented matrix An augmented matrix is a matrix that contains a submatrix that is the coefficient matrix of a particular system of linear equations, augmented with an additional column corresponding to the constant for each equation in the linear system. Given the system {, the corresponding augmented matrix is. Elementary Row Operations for Matrices 1. Replacement: add a multiple of one row to a second row to replace the second row 2. Scaling: multiply a row by a nonzero constant 3. Interchange: interchange any two rows In the above example, replacement was used on both Rows 2 and 3 to produce an equivalent system. In the above example, scaling was used on Row 3 to produce an equivalent system. In the above example, interchange was used on Rows 1 and 3 to produce an equivalent system.
3 Definition: row-echelon form (of a matrix) A matrix in row-echelon form is a matrix in which every entry beneath the main diagonal is zero. The matrices, and are all in row-echelon form. Definition: Gaussian elimination Gaussian elimination is an algorithm for solving systems of equations. It is accomplished by performing elementary row operations on the augmented matrix until the coefficient submatrix is in row-echelon form. Definition: reduced row-echelon form A matrix is in reduced row-echelon form if: 1. it is in row-echelon form 2. the first nonzero number in every row of the coefficient submatrix is 1 (known as a leading 1) 3. any column in the coefficient submatrix containing a leading 1 has all other column entries equal to zero The matrices and are in reduced row-echelon form. Determining the Intersection of Two Planes Along a Line Determine the nature of the intersection between the planes in the system {. The corresponding augmented matrix is and we can attempt to use elementary row operations to determine an equivalent matrix in row-echelon form. From Row 2 we can conclude that. Since we do not have a way of definitively determining particular values for and, we must parameterize one of our remaining variables. We will parameterize (although we could just as easily parameterize ). Let ( ). If, then (equation obtained from Row 1) implies that ( ) ( ) and thus. Thus the solution to this system is ( ) ( ). This implies ( ) ( ) ( ) which is the equation of a line. Thus the two planes intersect along this line.
4 unique solution to the linear system planes intersect at a point planes do not have collinear normal vectors planes intersect along a line planes do not have collinear normal vectors will need to parameterize a variable planes intersect along a line two planes are coincident the two coincident planes have collinear normal vectors will need to parameterize a variable all three planes intersect along a plane all three planes are coincident all three planes have collinear normal vectors could parameterize two variables, but unnecessarily no solutions to the linear system parallel planes both intersect the third plane along lines the two parallel planes have collinear normal vectors planes intersect pairwise along lines normal vectors are pairwise, not collinear two planes are coincident and the third is parallel and distinct the normal vectors to all three planes are collinear all three planes are parallel and distinct the normal vectors to all three planes are collinear the Cartesian equations are not proportional
5 Theorem: Cramer s Rule Cramer s Rule is a theorem that involves determining the solution to a linear system which has as many variables as equations, and has a unique solution. In Consider a linear system with a unique solution { which has an associated augmented matrix. The solution ( ) to the linear system can be uniquely determined using determinants defined: and. In Consider a linear system with a unique solution{ which has an associated augmented matrix. The solution ( ) to the linear system can be uniquely determined using determinants defined:,,.
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