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Linear Algebra For Dummies
Chapter 3: Mastering Matrices and Matrix Algebra . . . . . . . . . . . . . . .41
Getting Down and Dirty with Matrix Basics ...............................................41
Becoming familiar with matrix notation ...........................................42
Defining dimension ..............................................................................43
Putting Matrix Operations on the Schedule ...............................................43
Adding and subtracting matrices ......................................................43
Scaling the heights with scalar multiplication .................................45
Making matrix multiplication work ...................................................45
Putting Labels to the Types of Matrices .....................................................48
Identifying with identity matrices ......................................................49
Triangulating with triangular and diagonal matrices ......................51
Doubling it up with singular and non-singular matrices .................51
Connecting It All with Matrix Algebra .........................................................52
Delineating the properties under addition .......................................52
Tackling the properties under multiplication ..................................53
Distributing the wealth using matrix multiplication and addition 55
Transposing a matrix ..........................................................................55
Zeroing in on zero matrices ................................................................56
Establishing the properties of an invertible matrix ........................57
Investigating the Inverse of a Matrix ...........................................................58
Quickly quelling the 2 × 2 inverse ......................................................59
Finding inverses using row reduction ...............................................60
Chapter 4: Getting Systematic with Systems of Equations. . . . . . . . . .65
Investigating Solutions for Systems ............................................................65
Recognizing the characteristics of having just one solution .........66
Writing expressions for infinite solutions ........................................67
Graphing systems of two or three equations ...................................67
Dealing with Inconsistent Systems and No Solution .................................71
Solving Systems Algebraically .....................................................................72
Starting with a system of two equations ...........................................73
Extending the procedure to more than two equations ...................74
Revisiting Systems of Equations Using Matrices .......................................76
Instituting inverses to solve systems ................................................77
Introducing augmented matrices .......................................................78
Writing parametric solutions from augmented matrices ...............82
Part II: Relating Vectors and Linear Transformations .... 85
Chapter 5: Lining Up Linear Combinations . . . . . . . . . . . . . . . . . . . . . . .87
Defining Linear Combinations of Vectors ...................................................87
Writing vectors as sums of other vectors ........................................87
Determining whether a vector belongs .............................................89
Searching for patterns in linear combinations ................................93