Solutions of Linear system, vector and matrix equation

Transcription

1 Goals: Solutions of Linear system, vector and matrix equation Solutions of linear system. Vectors, vector equation. Matrix equation. Math 112, Week 2 Suggested Textbook Readings: Sections 1.3, 1.4, 1.5

2 Week 2: Solution set of Linear system, vector and matrix equation 2 Solve a linear system: 1. Write out the augmented matrix and row reduce to its Row Echelon Form; [ ] 2. If there is a row b, b 0, stop and conclude that the system has no solutions (i.e. inconsistent); 3. Otherwise, continue and get the RREF. 4. Write the system of equations corresponding to the RREF. 5. Read off the solutions, that is, rewrite each nonzero equation from step 4 so that the basic variable is expressed in terms of any free variables appearing in the equation. Basic variable: Free variable: Rank: Fact: Let A be an m n matrix, then rank(a) min{m, n}

3 Week 2: Solution set of Linear system, vector and matrix equation 3 Example 1: Let A and [A b] be the coefficient matrix and augmented matrix of a linear system, complete the table. size of A rank(a) rank[a b] Solution(s) number of free variables a unique solution Let A and [A b] be the coefficient matrix and augmented matrix of a linear system, then the linear system has: 1. no solutions if 2. a unique solution if 3. infinitely many solutions if

4 Week 2: Solution set of Linear system, vector and matrix equation 4 Vector: R n ={vectors with n entries} Example: vectors in R 2. equal: vector sum: scalar multiple: Geometric view of a vector in R 2 : point: arrow: Example 2: Plot the following vectors in the plane R 2 as arrows. v 1 = 1, v 2 = 1, v 3 = 1, v 4 = 1, v 5 =

5 Week 2: Solution set of Linear system, vector and matrix equation 5 Parallelogram rule for vector sum in R 2 : Scalar multiplication of a vector in R 2 : Example 3: Let u = 1, v = 1, find the vectors 2 1 in the following graph: u + 2v, 2u v, 2u v, 1 2 u 2v

6 Week 2: Solution set of Linear system, vector and matrix equation 6 vectors in R n : equal: vector sum: scalar multiple: Algebraic properties of R n : Linear combination. y = c 1 v 1 + c 2 v c p v p

7 Week 2: Solution set of Linear system, vector and matrix equation Example 4: Let a 1 = 2, a 2 = 3, b = Determine if b is a linear combination of a 1, a 2. Vector equation: x 1 a 1 + x 2 a x n a n = b

8 Week 2: Solution set of Linear system, vector and matrix equation 8 Span{v 1,, v p }: Fact: The zero vector 0 is in Span{v 1, v 2,, v p }. Example 5: Let v = 2, then Span{v}= 1 Example 6: Let u = 1, v = 1, then Span{u, v}= Example 7: Let u = 1, v = 2, then Span{u, v}= 1 1

9 Week 2: Solution set of Linear system, vector and matrix equation 9 Example 8: For what value(s) of h will b in Span{v 1, v 2, v 3 } if v 1 = 1, v 2 = 4, v 3 = 1, b = h The following statements are equivalent: b is a linear combination of v 1, v 2,, v p b is in Span{v 1,, v p }. The linear system with augmented matrix [v 1 v p b] has a solution.

10 Week 2: Solution set of Linear system, vector and matrix equation 10 Matrix equation Ax = b. matrix-vector product: Example 9: Compute x 1 x 2 x 3

11 Week 2: Solution set of Linear system, vector and matrix equation 11 Computation of Ax : Row-column rule. Example 10: Write the vector equation, and matrix equation of: { 3x1 +x 2 5x 3 = 4 x 2 + 4x 3 = 1 Theorem: Let A be an m n matrix with columns {a 1, a 2,, a n } in R m and b is in R m : matrix equation: vector equation: system of linear equations: They have the same solution set.

12 Week 2: Solution set of Linear system, vector and matrix equation 12 Let A denote an m n matrix, and a 1,, a n its column vectors, which are vectors in R m. Then the following statements are equivalent: 1. The vectors a 1,, a n generate R m (i.e. Span{a 1,, a n } = R m ) Linear system: Ax=b Homogeneous: Non-homogeneous:

13 Week 2: Solution set of Linear system, vector and matrix equation 13 Example 11: Solve the system and write the solutions in vector form. 3x 1 + 5x 2 4x 3 = 0 3x 1 2x 2 + 4x 3 = 0 6x 1 + x 2 8x 3 = 0 Example 12: Solve the system and write the solutions in vector form. 2x 1 3x 2 4x 3 = 0