ࡱ> `b _h[ ?bjbj %jj9ul...."NNN8 d"K-"8-8-.K8K8K8~KKKKKKK$L NKK85\K8K8K8KoD..8-.KoDoDoDK8d.8-.~KoDK8~KoDoDJVK@~K. s=^}",N8nK~KK0KvKO: O~KoD""....Linear Algebra Review For Abstract Algebra Neil Sigmon Contents 1. Matrices..page 1 2. Equality of Matrices. page 1 3. Special Types of Matrices Vectorspage 2 4. Addition, Subtraction and Scalar Multiplication of Matrices..page 2 5. Matrix Multiplicationpage 3 6. Addition and Multiplicative Identity Matrices..page 6 7. Upper Triangular and Diagonal Matrices..page 6 8. Determinants..page 7 9. Matrix Inverses..page 9 1. Matrices A matrix is a rectangular array of numbers (usually real) made up of rows and columns. The size of a matrix is its (# rows)  EMBED Equation.3 (# columns). Example 1: Here are some various matrices with different sizes:  EMBED Equation.3  (Size: 2 EMBED Equation.3 4)  EMBED Equation.3  (Size: 3 EMBED Equation.3 2)  EMBED Equation.3  (Size: 1 EMBED Equation.3 4)  EMBED Equation.3  (Size: 3 EMBED Equation.3 1) To indicate an individual entry in a matrix, we use the notation  EMBED Equation.3  (represents the element in the  EMBED Equation.3  row and  EMBED Equation.3 column of the matrix A) Thus, we have for  EMBED Equation.3  ( EMBED Equation.3 ) and for  EMBED Equation.3  ( EMBED Equation.3 ) A general matrix A of size  EMBED Equation.3  has the form  EMBED Equation.3  A square matrix is a matrix where (# rows) = (# columns) or m = n. When this is case, we say the matrix is a square matrix of order n. Here are some examples of square matrices:  EMBED Equation.3 (2 EMBED Equation.3  2 square matrix of order 2)  EMBED Equation.3 (3 EMBED Equation.3 3 square matrix of order 3). 2. Equality of Matrices Two matrices A and B are equal (written A = B) if they have the same size and their corresponding entries are equal, that is  EMBED Equation.3 for all i and j. Example 2: Given the matrices  EMBED Equation.3  and  EMBED Equation.3 , then A=C. However,  EMBED Equation.3  since these matrices do not have equal corresponding entries and  EMBED Equation.3  since the matrix sizes are not equal (A is a 2 EMBED Equation.3 3 matrix and D is a 3 EMBED Equation.3 2 matrix). 3. Special Types of Matrices Vectors A row vector is a matrix with one row. A column vector is a matrix with one column. Example 3: The matrix EMBED Equation.3  is a 1 EMBED Equation.3 4 row vector and the matrix EMBED Equation.3  is a 3 EMBED Equation.3 1 column vector. 4. Addition, Subtraction and Scalar Multiplication of Matrices Matrices can only be added and subtracted if they have the same size. To add or subtract matrices, we add or subtract each corresponding component. Example 4: Given the  EMBED Equation.3  matrices EMBED Equation.3  and  EMBED Equation.3 , we see that  EMBED Equation.3  and  EMBED Equation.3 . Example 5: Given the  EMBED Equation.3  matrix EMBED Equation.3  and the  EMBED Equation.3  matrix  EMBED Equation.3 , the matrix  EMBED Equation.3  is not defined since A and B are not of the same size. When working with matrices, numbers are referred to as scalars. To multiply a matrix by a scalar, we multiply each entry of the matrix by the given scalar. Example 6: Let  EMBED Equation.3 , then  EMBED Equation.3 . Example 7: Let  EMBED Equation.3 and  EMBED Equation.3 , then  EMBED Equation.3 . Matrix addition and scalar multiplication have certain basic properties that we now summarize Addition and Scalar Multiplication Properties of Matrices Let A, B, and C be matrices of the same size and c and d be scalars. 1. A + B = B + A (matrix addition is commutative) 2.  EMBED Equation.3  (matrix addition is associative) 3. cd(A) = c(dA). 4. c EMBED Equation.3 = c EMBED Equation.3 + c B 5. Matrix Multiplication To understand matrix multiplication, one must first understand how to multiply a row vector times a column vector. If  EMBED Equation.3  is a  EMBED Equation.3  row vector and  EMBED Equation.3  is a  EMBED Equation.3 column vector, then the product of A and B is the scalar produced by multiplying each corresponding entry of A and B and adding. That is  EMBED Equation.3 . Example 8: If  EMBED Equation.3  and  EMBED Equation.3 , then  EMBED Equation.3 . Multiplication of matrices in general involves multiple multiplications of rows and columns. If A is a  EMBED Equation.3 matrix and B is a  EMBED Equation.3 matrix, the product C=AB is the matrix where each element  EMBED Equation.3  is made up of the product of the  EMBED Equation.3  row of the left matrix A multiplied to the  EMBED Equation.3  column of the right matrix B. That is,  EMBED Equation.3 . Note! For the matrix product AB to exist, the number of columns of the left matrix A must equal to the number of rows in the right matrix B. The size of the product will be (the number of rows in A)  EMBED Equation.3 (the number of columns in B). It can be easier to see this by examining the following:  EMBED Equation.3   Example 9: If  EMBED Equation.3  and  EMBED Equation.3 , then the size (2  EMBED Equation.3 3) of the product AB is obtained by observing:  EMBED Equation.3   The 2  EMBED Equation.3 3 matrix AB is computing by performing the following row column multiplications:  EMBED Equation.3 . Performing these multiplications, we obtain:  EMBED Equation.3 . Example 10: For the matrices,  EMBED Equation.3 ,  EMBED Equation.3 , after observing from  EMBED Equation.3   that the size of the product is a  EMBED Equation.3  matrix, we obtain the product  EMBED Equation.3  by computing:  EMBED Equation.3 . To compute the product  EMBED Equation.3 , we observe  EMBED Equation.3   and compute  EMBED Equation.3 . Example 11: Consider the matrices  EMBED Equation.3  and  EMBED Equation.3 , one can immediately see by observing  EMBED Equation.3  that the product  EMBED Equation.3  does not exist (the number of columns in the left matrix A (4) is not equal to the number or rows in the right matrix B (3). However, by seeing  EMBED Equation.3   the product  EMBED Equation.3  is the  EMBED Equation.3  matrix given by  EMBED Equation.3 . The previous two examples illustrate a very important fact when multiplying matrices: FACT: In general, matrix multiplication is not commutative, that is, given matrices A and B, it is true in most cases that  EMBED Equation.3 . However, matrices that are multiplied do have several properties that are summarized: Multiplicative Properties of Matrices Let A, B, and C be matrices whose sizes are multiplicatively compatible, c a scalar. 1.  EMBED Equation.3  (matrix multiplication is associative) 2.  EMBED Equation.3 . 3.  EMBED Equation.3 . 4. c EMBED Equation.3 = (c EMBED Equation.3 B = A(cB) 6. Addition and Multiplicative Identity Matrices The zero matrix is a  EMBED Equation.3  matrix make up of all zeros. The following examples represent  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 zero matrices  EMBED Equation.3  The zero matrix represents the additive identity matrix of all matrices under the addition operation. If A is a  EMBED Equation.3  matrix and 0 is a  EMBED Equation.3  zero matrix, then A + 0 = 0 + A = A. The multiplicative identity matrix, usually referred to just the identity matrix I, is the  EMBED Equation.3  matrix defined by  EMBED Equation.3 . Note that I has 1s on the main diagonal and 0s elsewhere. The following represent a  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3  identity matrices:  EMBED Equation.3 . Since I serves as the multiplicative identity, if A is a  EMBED Equation.3  matrix, then AI = IA = A. Note that I is always a square matrix, that is, the number of rows equals the number of columns. Of course, the size of I is dependent on the size of A when multiplying on the left and right as the next example demonstrates. Example 12: Given the  EMBED Equation.3 matrix  EMBED Equation.3 , to be multiplicatively compatible, the size of the identity I on the left must be  EMBED Equation.3  and on the right  EMBED Equation.3 . This gives  EMBED Equation.3  and  EMBED Equation.3 . 7. Upper Triangular and Diagonal Matrices A square matrix is upper triangular if all entries below the main diagonal are zero. A diagonal matrix is a special case of an upper triangular matrix where entries both above and below the main diagonal are zero. The general form of each matrix looks like:  EMBED Equation.3   EMBED Equation.3  The matrix  EMBED Equation.3  is an example of a upper triangular matrix while  EMBED Equation.3  is an example of a diagonal matrix. Two important facts are evident when concerned with upper triangular and diagonal matrices: 1. The sum or difference of two upper triangular matrices is an upper triangular matrix and the sum or difference of two diagonal matrices is a diagonal matrix. 2. The product of two upper triangular matrices is an upper triangular matrix and the product of two diagonal matrices is a diagonal matrix. Example 13: Given the two upper triangular matrices  EMBED Equation.3  and  EMBED Equation.3  and two diagonal matrices  EMBED Equation.3  and  EMBED Equation.3 , one can readily verify the following sums and products:  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3  8. Determinants The determinant of a  EMBED Equation.3  matrix  EMBED Equation.3  is given by det(A)  EMBED Equation.3 . Example 14: The determinant of the matrix  EMBED Equation.3  is:  EMBED Equation.3  Example 15: The determinant of the matrix  EMBED Equation.3  is:  EMBED Equation.3  A fact to note is that the determinant of a  EMBED Equation.3  matrix is defined to be the entry of the matrix. For example, if  EMBED Equation.3 , then  EMBED Equation.3 . Another important note to make is that is possible to take determinants of larger size square matrices (larger than  EMBED Equation.3 ). However, this will not be necessary for most purposes except when we are dealing with diagonal and upper triangular matrices defined above. If we have an upper triangular and diagonal matrix of the form:  EMBED Equation.3  and  EMBED Equation.3 , then the determinant is defined to be the product of the main diagonal elements, that is  EMBED Equation.3  and  EMBED Equation.3 . Example 16: Given the upper triangular matrix  EMBED Equation.3  and diagonal matrix  EMBED Equation.3 , we see that  EMBED Equation.3  and  EMBED Equation.3 . An important property concerning the determinant of the product of two matrices is the following:  EMBED Equation.3  Example 17: For the matrices  EMBED Equation.3  and  EMBED Equation.3 , one can verify that  EMBED Equation.3  and compute  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Thus one can see that  EMBED Equation.3 . However, this distributive property does not in general hold for the sum of two matrices, that is, in general:  EMBED Equation.3  Example 18: For the matrices  EMBED Equation.3  and  EMBED Equation.3 , one can verify that  EMBED Equation.3  and compute  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Thus one can see that  EMBED Equation.3 . 9. Matrix Inverses Matrices defined with addition have an obvious inverse. Given a matrix A, its additive inverse is defined to be the matrix A, where all the elements of A are negated. Note that A + (-A) = 0, where 0 is the zero matrix. We are more interested however with the multiplicative inverse of a matrix, or inverse for short. The inverse of a  EMBED Equation.3  matrix  EMBED Equation.3 , if it exists, is denoted by  EMBED Equation.3 , and is defined to be the  EMBED Equation.3  matrix where  EMBED Equation.3 , where  EMBED Equation.3  is the  EMBED Equation.3  identity matrix. It can be shown that  EMBED Equation.3  is unique. Note the inverse only exists for square matrices where the row and column number are the same. Example 19: To show that  EMBED Equation.3  is the inverse of the matrix  EMBED Equation.3 , one computes  EMBED Equation.3 . A similar calculation can be shown for  EMBED Equation.3 . Given a matrix  EMBED Equation.3 , how do we know if  EMBED Equation.3  exists, and if it does, how can we calculate  EMBED Equation.3 ? For matrices larger than  EMBED Equation.3 , the recommended method involves a method involving row reduction. However, for  EMBED Equation.3  matrices, there is a method based upon the following: FACT: For a given  EMBED Equation.3  matrix EMBED Equation.3 , the matrix  EMBED Equation.3  exists only if  EMBED Equation.3 . If this is so, the inverse of the matrix is defined to be:  EMBED Equation.3 . Note that the matrix part of the formula for  EMBED Equation.3  is obtained by switching the main diagonal elements and negating the back diagonal elements. Example 20: Given the matrix  EMBED Equation.3  we see that  EMBED Equation.3 . Hence the inverse exists and  EMBED Equation.3  . Example 21: Given the matrix EMBED Equation.3 , one can quickly see that  EMBED Equation.3 . Therefore,  EMBED Equation.3  does not exist. Example 22: Given the diagonal matrix  EMBED Equation.3 , one can see that  EMBED Equation.3 . Thus  EMBED Equation.3  does not exist. For the upper triangular matrix  EMBED Equation.3 , one can see that  EMBED Equation.3 . Thus  EMBED Equation.3  exists, although row reduction methods are needed to verify that  EMBED Equation.3 . We end with two more useful facts concerning inverses without verification: 1. The inverse of a diagonal matrix is diagonal. 2. The inverse of an upper triangular matrix is upper triangular. 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