of Multiplication a Matrix by a Vector
Multiplication of a Matrix by a Vector
Let A be an m n matrix, with columns a1 , a2 ,. . ., an which are vectors in Rm Let x be a vector in Rn . The product of A and x, written Ax, is the vector in Rm that is given by the linear combination of the columns of A using the corresponding entries in x as weights, that is
Ax = a1 a2
an
x1 x2 . . . xn
= x1 a1 + x2 a2 + + xn an
Clint Lee
Math 221 Lecture 4: Matrix Equations
2/13
The Row-Vector Rule
The Row-Vector Rule
The i th entry of the product Ax is the sum of the of the products of corresponding entries from the i th row of A and from the vector x, that is if
A=
then
a11 a22 . . .
a12 a22 . . .
.. .
a1n a2n . . .
and x =
x1 x2 . . . xn
[Ax]i = x1 ai1 + x2 ai2 + + xn ain
This is the row-vector rule for computing Ax.
Clint Lee
Math 221 Lecture 4: Matrix Equations
3/13
Not All Matrix-Vector Products Can Be Formed
Not All Matrix-Vector Products Can Be Formed
Not all matrix/vector products can be formed. The number of columns in matrix A must equal the number of entries in the vector x. Example 12 Problem 3 page 47
Clint Lee
Math 221 Lecture 4: Matrix Equations
4/13
Matrix Equations
The matrix equation, Ax = b, states that the vector b can be written as linear combination of the columns of the matrix A with weights equal to the entries of some unknown vector x. Example 13 Problem 7 page 47
Clint Lee
Math 221 Lecture 4: Matrix Equations
5/13
Matrix Equations, Vector Equations, and Linear Systems
A matrix equation can always be written as a vector equation or a linear system or vice versa. This is stated in
Theorem (1.3)
If A is an m n matrix, with columns a1 , a2 ,. . . , an , and if b is a vector in Rm the matrix equation Ax = b is equivalent to the vector equation x1 a1 + x2 a2 + + xn an = b which in turn has the same solution set as the linear system with augmented matrix [a1 a2 . . . an b]
Clint Lee
Math 221 Lecture 4: Matrix Equations
6/13
Comments on Theorem 1.3
Comments on Theorem 1.3
The matrix equation Ax = b has a solution if and only if the vector b is a linear combination of the columns of matrix A The matrix equation Ax = b has a solution and if and only if the linear system with augmented matrix
[a1 a2 . . . an b]
is consistent.
Example 14 Problem 11 page 47 Example 15 Problem 15 page 48
Clint Lee
Math 221 Lecture 4: Matrix Equations
7/13
When Does Ax = b Have a Solution for All b
Theorem (1.4)
Let A be an m n matrix. Then the following statements are logically equivalent, i.e., for a given matrix A they are all true statements or they are all a. false. 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 matrix A span Rm . d. A has pivot positions in every row.
Example 16 Problems 17 and 19 page 48 Example 17 Problem 21 page 48
Clint Lee
Math 221 Lecture 4: Matrix Equations
8/13
Proof of Theorem 1.4
Proof of Theorem 1.4
a. and b. are equivalent by Theorem 1.3 c. is simply another way to say that every b in Rm is a linear combination of the columns of A Let U be an echelon form of the matrix A For any b in Rm the augmented matrix [A b] can be row reduced to [U d] where d is some vector in Rm If d. is true, then U has a pivot in each row so that the last column of [U d] cannot be a pivot column. Hence the system Ax = b has a solution for any b, i.e., a. is true.
Clint Lee
Math 221 Lecture 4: Matrix Equations
9/13
Proof of Theorem 1.4 continued
Proof of Theorem 1.4 continued
If d. is false, then at least one row in U does not have a pivot. If a row does not contain a pivot it must contain all zeros. Thus U has at least one row that is all zeros which will be its last row. If d is a vector in Rm whose last entry is 1, the augmented matrix [U d], whose last column is a pivot column, can be transformed to [A b] (row operations are reversible, and U and A are row equivalent), where b is some vector in Rm . Since an echelon form of the augmented matrix [A b] has the last column that is a pivot column, the system Ax = b is inconsistent, i.e., a. is false by Theorem 2.
Clint Lee
Math 221 Lecture 4: Matrix Equations
10/13
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