Linear Algebra (chapter1)02
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§1.4 The Matrix Equation Ax=b
1. Definition 2. Existence of Solutions 3. Computation of Ax 4. Properties of the Matrix-Vector Product Ax
§1.4 The Matrix Equation Ax=b
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions
If the equation Ax=b is consistent, b must satisfy:
This is the equation of a plane through the origin in R3:
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions
Sol. Row reduce the augmented matrix for Ax=b:
∵ ∴
=
≠ 0 (for some choices of b)
ห้องสมุดไป่ตู้
The equation Ax=b is not consistent for every b.
§1.5 Solution Set of Linear Systems
Use vector notation to describe solutions sets of linear system 1. Homogeneous Linear Systems （齐次线性方程组） 2. Parametric Vector Form 3. Solutions of Nonhomogeneous Systems
⎡ 4 ⎤ ⎡ 6 ⎤ ⎡ − 7 ⎤ ⎡ 3⎤ =⎢ ⎥+⎢ ⎥ + ⎢ 21 ⎥ = ⎢6⎥ ⎣0⎦ ⎣− 15⎦ ⎣ ⎦ ⎣ ⎦
b.
⎡ 2 − 3⎤ ⎡ 2 ⎤ ⎡− 3⎤ ⎡− 13⎤ ⎢8 ⎥ ⎡4⎤ = 4⎢ 8 ⎥ + 7 ⎢ 0 ⎥ = ⎢ 32 ⎥ 0 ⎥⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ 7⎦ ⎢ − 5⎥ ⎢ 2 ⎥ ⎢ − 6 ⎥ ⎢− 5 2 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎣ ⎦
1.Definition Definition of Ax
If A is an m×n matrix, with column a1,…,an, and if x is in Rn, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is:
Matrix equation
⎡ x1 ⎤ ⎡1 2 −1⎤ ⎢ ⎥ ⎡ 4 ⎤ ⎢0 −5 3 ⎥ ⎢ x2 ⎥ = ⎢1 ⎥ ⎣ ⎦⎢ ⎥ ⎣ ⎦ ⎣ x3 ⎦
They are all solved in the same way ___ by row reducing the augmented matrix.
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems Example1
x1 + 10 x2 = 0 2 x1 + 20 x2 = 0
Sol. (1)
Corresponding matrix equations
⎡1 10 ⎤ ⎡ x1 ⎤ ⎢2 20⎥ ⎢ x ⎥ = 0 ⎣ ⎦⎣ 2 ⎦
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems
(2) Row reduction to reduced echelon form
(3) The general solution
§1.5 Solution Set of Linear Systems
§1.4 The Matrix Equation Ax=b
3. Computation of Ax Example 5
Ix=x
§1.4 The Matrix Equation Ax=b
4. Properties of the Matrix-Vector Product Ax Theorem 5 If A is an m×n matrix, u and v are vectors in Rn, and c is a scalar, then: a. A(u+v)=Au+Av b. A(cu)=c(Au)
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems Homogeneous Linear Systems trivial solution (平凡解) x = 0, zero solution Ax = 0
nontrivial solution (非平凡解) a nonzero vector solution
The columns of A=[a1 a2 a3] span a plane through 0.
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions Theorem 4
Let A be an m×n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. a. For each b in Rm, the equation Ax = b has a solution. b. Each b in Rm is a linear combination of the columns of A. c. The columns of A span Rm. d. A has a pivot position in every row. [A is a coefficient, not augmented matrix]
§1.4 The Matrix Equation Ax=b
1.Definition Example 1
a.
⎡ 4⎤ ⎡1⎤ ⎡ 2 ⎤ ⎡− 1⎤ ⎡1 2 − 1⎤ ⎢ ⎥ ⎢0 − 5 3 ⎥ ⎢3⎥ = 4 ⎢0⎥ + 3⎢− 5⎥ + 7 ⎢ 3 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎦ ⎢7 ⎥ ⎣ ⎣ ⎦
which, in turn, has the same solution set as the system of linear equation whose augmented matrix is
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions
Row-vector rule for computing Ax If the product Ax is defined ,then the ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector
⎡0 ⎤ x=⎢ ⎥ ⎣0 ⎦
or
x=0
The homogeneous system Ax=0 always have the trivial system solution x=0
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems
The first entry of Ax is a sum of the products of corresponding entries from row 1 of A and from the vector x.
⎡ 2 3 4 ⎤ ⎡ x1 ⎤ ⎡ 2 x1 + 3 x2 + 4 x3 ⎤ ⎢ ⎥ ⎢x ⎥ = ⎢ ⎥ ⎢ ⎥⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x3 ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦
Fact
The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems Example2 Determine if the homogeneous system has a
nontrivial equation. Then describe the solution set.
Sol. (1) Row reduction
§1.4 The Matrix Equation Ax=b
1.Definition Example 2
For v1,v2,v3 in Rm, write the linear combination as a matrix times a vector.
Sol.
§1.4 The Matrix Equation Ax=b
fact The equation Ax=b has a solution if and only if b is a linear combination of columns of A.
Example3. Is the equation Ax=b consistent for all possible b1,b2,b3?
§1.4 The Matrix Equation Ax=b
1.Definition Theorem 3
If A is an m×n matrix, with column a1,…,an, and if b is in Rm, the matrix equation Ax = b has the same solution set as the vector equation
§1.4 The Matrix Equation Ax=b
3. Computation of Ax A more efficient method for Ax Example 4 Compute Ax, where
Sol.
§1.4 The Matrix Equation Ax=b
3. Computation of Ax
2. Parametric vector form Implicit description
10 x1 − 3 x2 − 2 x3 = 0
explicit description
⎡ x1 ⎤ ⎡.3 x2 + .2 x3 ⎤ ⎡.3⎤ ⎡.2 ⎤ ⎥ = x ⎢1⎥ + x ⎢0⎥ x = ⎢ x2 ⎥ = ⎢ x2 2 ⎢ ⎥ 3⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x3 ⎥ ⎢ ⎥ ⎢0⎥ ⎢1⎥ x3 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
1.Definition
Three different but equivalent ways: System of linear equations
x1 + 2 x2 − x3 = 4 − 5 x2 + 3 x3 = 1
Vector equation
⎡1 ⎤ ⎡2⎤ ⎡ −1⎤ ⎡ 4 ⎤ x1 ⎢ ⎥ + x2 ⎢ ⎥ + x3 ⎢ ⎥ = ⎢ ⎥ ⎣0 ⎦ ⎣ −5⎦ ⎣ 3 ⎦ ⎣1 ⎦
(2) Is there exists a nontrivial solutions?
x1 + 10 x2 = 0 2 x1 + 20 x2 = 0
⎡ 1 10 0 ⎤ ⎡1 10 0⎤ ⎢2 20 0⎥ ~ ⎢ 0 0 0 ⎥ ⎣ ⎦ ⎣ ⎦
Consistent system with a free variable has infinitely many solutions
1. Definition 2. Existence of Solutions 3. Computation of Ax 4. Properties of the Matrix-Vector Product Ax
§1.4 The Matrix Equation Ax=b
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions
If the equation Ax=b is consistent, b must satisfy:
This is the equation of a plane through the origin in R3:
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions
Sol. Row reduce the augmented matrix for Ax=b:
∵ ∴
=
≠ 0 (for some choices of b)
ห้องสมุดไป่ตู้
The equation Ax=b is not consistent for every b.
§1.5 Solution Set of Linear Systems
Use vector notation to describe solutions sets of linear system 1. Homogeneous Linear Systems （齐次线性方程组） 2. Parametric Vector Form 3. Solutions of Nonhomogeneous Systems
⎡ 4 ⎤ ⎡ 6 ⎤ ⎡ − 7 ⎤ ⎡ 3⎤ =⎢ ⎥+⎢ ⎥ + ⎢ 21 ⎥ = ⎢6⎥ ⎣0⎦ ⎣− 15⎦ ⎣ ⎦ ⎣ ⎦
b.
⎡ 2 − 3⎤ ⎡ 2 ⎤ ⎡− 3⎤ ⎡− 13⎤ ⎢8 ⎥ ⎡4⎤ = 4⎢ 8 ⎥ + 7 ⎢ 0 ⎥ = ⎢ 32 ⎥ 0 ⎥⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ 7⎦ ⎢ − 5⎥ ⎢ 2 ⎥ ⎢ − 6 ⎥ ⎢− 5 2 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎣ ⎦
1.Definition Definition of Ax
If A is an m×n matrix, with column a1,…,an, and if x is in Rn, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is:
Matrix equation
⎡ x1 ⎤ ⎡1 2 −1⎤ ⎢ ⎥ ⎡ 4 ⎤ ⎢0 −5 3 ⎥ ⎢ x2 ⎥ = ⎢1 ⎥ ⎣ ⎦⎢ ⎥ ⎣ ⎦ ⎣ x3 ⎦
They are all solved in the same way ___ by row reducing the augmented matrix.
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems Example1
x1 + 10 x2 = 0 2 x1 + 20 x2 = 0
Sol. (1)
Corresponding matrix equations
⎡1 10 ⎤ ⎡ x1 ⎤ ⎢2 20⎥ ⎢ x ⎥ = 0 ⎣ ⎦⎣ 2 ⎦
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems
(2) Row reduction to reduced echelon form
(3) The general solution
§1.5 Solution Set of Linear Systems
§1.4 The Matrix Equation Ax=b
3. Computation of Ax Example 5
Ix=x
§1.4 The Matrix Equation Ax=b
4. Properties of the Matrix-Vector Product Ax Theorem 5 If A is an m×n matrix, u and v are vectors in Rn, and c is a scalar, then: a. A(u+v)=Au+Av b. A(cu)=c(Au)
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems Homogeneous Linear Systems trivial solution (平凡解) x = 0, zero solution Ax = 0
nontrivial solution (非平凡解) a nonzero vector solution
The columns of A=[a1 a2 a3] span a plane through 0.
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions Theorem 4
Let A be an m×n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. a. For each b in Rm, the equation Ax = b has a solution. b. Each b in Rm is a linear combination of the columns of A. c. The columns of A span Rm. d. A has a pivot position in every row. [A is a coefficient, not augmented matrix]
§1.4 The Matrix Equation Ax=b
1.Definition Example 1
a.
⎡ 4⎤ ⎡1⎤ ⎡ 2 ⎤ ⎡− 1⎤ ⎡1 2 − 1⎤ ⎢ ⎥ ⎢0 − 5 3 ⎥ ⎢3⎥ = 4 ⎢0⎥ + 3⎢− 5⎥ + 7 ⎢ 3 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎦ ⎢7 ⎥ ⎣ ⎣ ⎦
which, in turn, has the same solution set as the system of linear equation whose augmented matrix is
§1.4 The Matrix Equation Ax=b
2. Existence of Solutions
Row-vector rule for computing Ax If the product Ax is defined ,then the ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector
⎡0 ⎤ x=⎢ ⎥ ⎣0 ⎦
or
x=0
The homogeneous system Ax=0 always have the trivial system solution x=0
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems
The first entry of Ax is a sum of the products of corresponding entries from row 1 of A and from the vector x.
⎡ 2 3 4 ⎤ ⎡ x1 ⎤ ⎡ 2 x1 + 3 x2 + 4 x3 ⎤ ⎢ ⎥ ⎢x ⎥ = ⎢ ⎥ ⎢ ⎥⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x3 ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦
Fact
The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.
§1.5 Solution Set of Linear Systems
1. Homogeneous Linear Systems Example2 Determine if the homogeneous system has a
nontrivial equation. Then describe the solution set.
Sol. (1) Row reduction
§1.4 The Matrix Equation Ax=b
1.Definition Example 2
For v1,v2,v3 in Rm, write the linear combination as a matrix times a vector.
Sol.
§1.4 The Matrix Equation Ax=b
fact The equation Ax=b has a solution if and only if b is a linear combination of columns of A.
Example3. Is the equation Ax=b consistent for all possible b1,b2,b3?
§1.4 The Matrix Equation Ax=b
1.Definition Theorem 3
If A is an m×n matrix, with column a1,…,an, and if b is in Rm, the matrix equation Ax = b has the same solution set as the vector equation
§1.4 The Matrix Equation Ax=b
3. Computation of Ax A more efficient method for Ax Example 4 Compute Ax, where
Sol.
§1.4 The Matrix Equation Ax=b
3. Computation of Ax
2. Parametric vector form Implicit description
10 x1 − 3 x2 − 2 x3 = 0
explicit description
⎡ x1 ⎤ ⎡.3 x2 + .2 x3 ⎤ ⎡.3⎤ ⎡.2 ⎤ ⎥ = x ⎢1⎥ + x ⎢0⎥ x = ⎢ x2 ⎥ = ⎢ x2 2 ⎢ ⎥ 3⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x3 ⎥ ⎢ ⎥ ⎢0⎥ ⎢1⎥ x3 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
1.Definition
Three different but equivalent ways: System of linear equations
x1 + 2 x2 − x3 = 4 − 5 x2 + 3 x3 = 1
Vector equation
⎡1 ⎤ ⎡2⎤ ⎡ −1⎤ ⎡ 4 ⎤ x1 ⎢ ⎥ + x2 ⎢ ⎥ + x3 ⎢ ⎥ = ⎢ ⎥ ⎣0 ⎦ ⎣ −5⎦ ⎣ 3 ⎦ ⎣1 ⎦
(2) Is there exists a nontrivial solutions?
x1 + 10 x2 = 0 2 x1 + 20 x2 = 0
⎡ 1 10 0 ⎤ ⎡1 10 0⎤ ⎢2 20 0⎥ ~ ⎢ 0 0 0 ⎥ ⎣ ⎦ ⎣ ⎦
Consistent system with a free variable has infinitely many solutions