Lecture 5 Notes

System of equations x1 2x2 4 3x1 4x2 7 operator

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Unformatted text preview: e connections to linear (matrix) algebra • An mxn matrix is a gizmo that takes an n-vector and returns an mvector: y = Ax Some connections to linear (matrix) algebra • An mxn matrix is a gizmo that takes an n-vector and returns an mvector: y = Ax • It is called a linear operator because it has the following properties: A(cx) = cAx A(x + y ) = Ax + Ay Some connections to linear (matrix) algebra • An mxn matrix is a gizmo that takes an n-vector and returns an mvector: y = Ax • It is called a linear operator because it has the following properties: A(cx) = cAx A(x + y ) = Ax + Ay • Not all operators work on vectors. Derivative operators take a function and return a new function. For example, d2 y dy z = L[y ] = 2 − 2 +y dt dt Some connections to linear (matrix) algebra • An mxn matrix is a gizmo that takes an n-vector and returns an mvector: y = Ax • It is called a linear operator because it has the following properties: A(cx) = cAx A(x + y ) = Ax + Ay • Not all operators work on vectors. Derivative operators take a function and return a new function. For example, d2 y dy z = L[y ] = 2 − 2 +y dt dt • This one is linear because L[cy ] = cL[y ] L[y + z ] = L[y ] + L[z ] Note: y, z are functions of t and c is a constant. Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form Ax = 0 Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form Ax = 0 • A non-homogeneous matrix equation has the form Ax = b Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form Ax = 0 • A non-homogeneous matrix equation has the form Ax = b • A homogeneous differential equation has the form L[y ] = 0 Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form Ax = 0 • A non-homogeneous matrix equation has the form Ax = b • A homogeneous differential equation has the form L[y ] = 0 • A non-homogeneous differential equation has the form L[y ] = g (t) Some connections to linear (matrix) algebra Systems of equations written in operator notation. System of equations Operator deﬁnition Equation in operator notation Some connections to linear (matrix) algebra Systems of equations written in operator notation. System of equations x1 + 2x2 = 4 3x1 + 4x2 = 7 Operator deﬁnition Equation in operator notation Some connections to linear (matrix) algebra Systems of equations written in operator notation. System of equations x1 + 2x2 = 4 3x1 + 4x2 = 7 Operator deﬁnition Ax = ￿ 1 3 ￿￿ ￿ 2 x1 4 x2 Equation in operator notation Some connections to linear (matrix) algebra Systems of equations written in operator notation. System of equations x1 + 2x2 = 4 3x1 + 4x2 = 7 Operator deﬁnition Ax = ￿ 1 3 ￿￿ ￿ 2 x1 4 x2 Equation in operator notation ￿￿ 4 Ax = 7 Some connections to linear (matrix) algebra Systems of equations written in operator notation. System of equations x1 + 2x2 =...
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This note was uploaded on 02/12/2014 for the course MATH 256 taught by Professor Ericcytrynbaum during the Spring '13 term at UBC.

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