Lecture 4 Lines in R Systems of linear equations as you change c you are translating the parallel lines There are in nitely man equations for each line given by multiplying through by scalar The solution is the intersection of these two lines There are no solutions if the lines are parallel they never intersect There are in nitely many solutions if the lines are parallel The lines are parallel iff det 0 Matrices and linear equations We can write the system of the equations using matrices If the determinate of A 0 then the inverse of A exists Recall A A A A I If A then det A ad bc We showed that if the determinant does not equal zero there i s exactly one solution Then A is mxm and the determinant of A does not equal zero Then Ax c has exactly one solution x Ac Systems of equations in R System of equations Matrix equation Homogeneous systems Example If the determinant of A does not equal zero there is exactly one solution If the determinant of A equals zero there can be many or no solutions
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