Lecture 6 Notes

2 13 23 53 23 0 0 1 2 5 2 x1 x3 and x2 x3 3 3 3 3 and

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Unformatted text preview: Ax = b 3 −2 1 and where 2 b = 0 . 2 Solutions to non-homogeneous matrix equations • Example 3. Solve the equation 1 A = 1 2 • Row reduction gives 2 −1 1 Ax = b 3 −2 1 10 0 1 00 and where 2 b = 0 . 2 ￿ −1/3￿ 2/3 ￿ 5/3 ￿ 2/3 ￿ 0 ￿0 Solutions to non-homogeneous matrix equations • Example 3. Solve the equation 1 A = 1 2 • Row reduction gives • so 2 −1 1 Ax = b 3 −2 1 10 0 1 00 and where 2 b = 0 . 2 ￿ −1/3￿ 2/3 ￿ 5/3 ￿ 2/3 ￿ 0 ￿0 1 2 5 2 x1 − x3 = and x2 + x3 = 3 3 3 3 and x3 can be whatever. Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = 3 3 • Example 3. Solve the equation • so 1 2 x1 − x3 = and 3 3 and x3 can be whatever. Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 and x3 can be whatever. Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 1 2/3 C −5 + 2/3 x= 3 3 0 Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem one particular solution to nonhomogeneous problem Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem one particular solution to nonhomogeneous problem Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem one particular solution to nonhomogeneous problem Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem one particular solution to nonhomogeneous problem Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem one particular solution to nonhomogeneous problem Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem one particular solution to nonhomogeneous problem Solutions to non-homogeneous matrix equations Ax = b . 5 2 x2 + x3 = and x3 can be whatever. 3 3 5 2 x2 = − x3 + 3 3 • Example 3. Solve the equation 1 • so x1 − x3 = 3 1 x1 = x3 + 3 2 and 3 2 3 2/3 1 ￿ x = C −5 + 2/3 0 3 the general solution to the homogeneous problem one particular solution to nonhomogeneous problem Solutions to non-homogeneous matrix eq...
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