线性代数课件2

Unique Solution For 3 × 2 Systems
x2 dx1 + ex 2 = f gx1 + hx 2 = i ax1 + bx 2 = c x1
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Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
Case II: Infinitely Many Solutions
For a 2 × 2 system:
x2 ax1 + bx2 = c dx1 + ex2 = f
x2 ax1 + bx 2 = c dx1 + ex 2 = f gx1 + hx 2 = i ! ! ! x1
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Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
x1
This happens when lines are on top of each other (fully or partially) d= k· a e= k· b f = k · c for some k.
Solution Sets
Equivalent Systems
Gaussian Elimination
x1 dx1 + ex 2 = f
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
Infinitely Many Solutions For 3 × 2 Systems
Lecture 2 Solving Linear Equations Using Elimination Techniques
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
Gauss-Jordan Elimination
Matrix Representation
Case I: No Solution
For a 2 × 2 system:
x2 ax1 + bx2 = c dx1 + ex2 = f
x1
This happens when there is no overlap (lines don’t intersect) d= k· a e= k· b f = k· c for some k
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
Scaling
The scaling operation is: cEi → Ei This means “multiply each side of the ith equation by c and replace the ith equation with what you obtain.” Example: If c = 2, 2x1 + 3x2 = 5 becomes 4x1 + 6x2 = 10 Since this is really the same equation, scaling does not change the solution set.
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
EE 20: Linear Methods for Engineering Analysis and Design Using MATLAB
Matrix Representation
Elementary Equation Operations (EEOs)
If one performs one of the three operations
scaling, interchange, or replacement,
then the linear system changes, but the solution set remains the same. Then the initial and the final systems are said to be equivalent. We will also observe that each of the three operations is reversible.
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
No Solution For 3 × 2 Systems
x2 gx1 + hx 2 = i ax1 + bx 2 = c ! ! ! ! x1 x2 ax1 + bx 2 = c ! ! ! ! gx1 + hx 2 = i x1 ! ! ! x1 dx1 + ex 2 = f ! x2 ! ! ax1 + bx 2 = c gx1 + hx 2 = i dx1 + ex 2 = f ! ! ! ! x2 gx1 + hx 2 = i ax1 + bx 2 = c dx1 + ex 2 = f
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
Interchange
The interchange operation is: Ei ↔ Ej
This means “swap the ith and jth equations,” which clearly does not change the solution set either. Example: If the system is 2x1 + 3x2 − 4x3 = 1 −3x1 − 5x2 + 8x3 = −16 5x1 + 314x3 = 18 then E1 ↔ E2 changes it to −3x1 − 5x2 + 8x3 = −16 2x1 + 3x2 − 4x3 = 1 5x1 + 314x3 = 18
Linear Equation Systems
A linear system looks like: 2x1 + 3x2 − 4x3 = 1 −3x1 − 5x2 + 8x3 = −16 5x1 + 314x3 = 18 In general, there are m equations and n unknowns (m × n system) Definition: The solution set is the set of (x1 , x2 , . . . , xn ) that satisfy each one of the m equations simultaneously. Solving the system means determining the solution set. Three possibilities:
it satisfies both Ei and Ej =⇒ it also satisfies aEi + bEj =⇒ it is also a member of the new solution set.
Conversely, if (x1 , x2 , . . . , xn ) is a member of the new solution set,
Gauss-Jordan Elimination
Matrix Representation
Case III: Unique Solution
For a 2 × 2 system:
x2 dx1 + ex2 = f
ax1 + bx2 = c x1
This happens when lines intersect. (All other cases) Examples: /algebra/linear_equation/systems-of-equation/real-world-application.php
Solution Sets
Equivalent Systems
Gaussian Elimination
Gauss-Jordan Elimination
Matrix Representation
Forward Elimination
Step 1: Apply aE1 + Ei → Ei to all i > 1 so as to eliminate x1 from all the equations below E1 . Example: 2x1 + 3x2 − 4x3 = 1 −3x1 − 5x2 + 8x3 = −16 5x1 + 314x3 = 18 Apply 1.5E1 + E2 → E2 : 2x1 + 3x2 − 4x3 = 1 − 0.5x2 + 2x3 = −14.5 5x1 + 314x3 = 18 2x1 + 3x2 − 4x3 = 1 − 0.5x2 + 2x3 = −14.5 − 7.5x2 + 324x3 = 15.5