Calculus

A function is a rule that assigns each element in a set D to exactly one element in a set R.

If f is an even function, then f(-x) = f(x) for all x. If f is an odd function, then f(-x) = -f(x) for all x. f(x) = x^2 is an even function because f(-x) = (-x)^2 = (-x)(-x) = x^2 = f(x). f(x) = x^3 is an odd function because f(-x) = (-x)^3 = (-x)(-x)(-x) = (-x)x^2 = -(x^3) = -f(x). Odd functions are symmetric with respect to the origin. Even functions are symmetric with respect to the y-axis.

A function is called increasing on an interval I if for all x1 < x2 in I, f(x1) < f(x2). And a function is called decreasing on an interval I if for all x1 < x2 in I, f(x1) > f(x2).

Linear Algebra (5)

Ax = 0 is a homogeneous system of linear equations. 0 represents the zero vector. Every homogeneous system of linear equations has at least one solution which is the trivial solution or in other words the zero vector. A homogeneous system of linear equations with a nontrivial solution can be thought of as a set of vectors that when summed together start at the origin and end at the origin. Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

Linear Algebra (4)

A vector y is said to be a linear combination of a set of vectors v1, v2, ... , vn if there exists scalars c1, c2, ... , cn, such that y = c1(v1) + c2(v2) + ... + cn(vn). span{v1, v2, ... , vn} is the set of all linear combinations of v1, v2, ... , vn. A linear combination can be viewed as a product of a matrix and a vector. The vector is the set of scalars and the matrix the set of vectors. The product of a matrix A and a vector x is written Ax. Ax = b has a solution if and only if b is a linear combination of the columns of A.

                                  

Ax = [ a1 , a2 , ... , an ]x =  x1(a1) + x2(a2) + ... + xn(an)

So, linear systems can be viewed in three different ways. As a matrix equation, a vector equation, or as a set of linear equations in an augmented matrix.

A very important matrix to understand is the identity matrix. It is any square matrix with 1's along the diagonal and zeros elsewhere. 

1   0   0   0   0   0
0   1   0   0   0   0
0   0   1   0   0   0
0   0   0   1   0   0
0   0   0   0   1   0
0   0   0   0   0   1

is a 6 x 6 identity matrix.

If A is an m x n matrix, u and v are vectors in ℝ^n, and c is a scalar, then A(u + v) = Au + Av and A(cu) = c(Au).


                                         

Linear Algebra (3)

A matrix with only one column is called a vector.

         1
V =  2
         3

V can also be expressed as V = [ 1 , 2 , 3 ]. Essentially, a vector is an ordered set of numbers. The set of all vectors with three entries is denoted by  ℝ^3. Given two vectors u and v in ℝ^3, their sum u + v is a vector w obtained by adding the corresponding entries of u and v. For example, if u = [ 1 , 2 , 3 ] and v = [ 2 , 3 , 4 ], then w = u + v = [ 1 , 2 , 3 ] + [ 2 , 3 , 4 ] = [ 1 + 2 , 2 + 3 , 3 + 4 ] = [ 3 , 5 , 7 ].

Given a vector u and a real number c, the scalar multiple of u by c is the vector c(u) obtained my multiplying each entry in u by c. The number c in c(u) is called a scalar.

Geometric Descriptions of ℝ^2
The following is a graph of the set of all vectors in ℤ^2 which is a subset of ℝ^2 such that each vector is not a scalar multiple of any other vector in the set and their components are between -8 to 8. The vectors that are in the set are marked with a purple dot. 

Vector addition geometrically amounts to the construction of a parallelogram.

Here are some algebraic properties of ℝ^n:

For all u, v, and w in ℝ^n and scalars c and d

1.) u + v = v + u

2.) (u + v) + w = u + (v + w)

3.) u + 0 = u

4.) u + (-u) = 0

5.) c(u + v) = cu + cv

6.) (c + d)u = cu + du

7.) c(du) = (cd)u

8.) 1u = u

Linear Algebra (2)

In a matrix a leading entry is the leftmost nonzero entry in a row. Consider the following matrix.

1   3   4   5   6
0   1   3   5   7
0   0   1   2   3

Rows 1, 2, and 3 have leading entries at columns 1, 2, and 3 respectively. A matrix is in row echelon form if all nonzero rows are above any rows of all zeros, each leading entry of a row is in a column to the right of the leading entry of the row above it, and all entries in a column below a leading entry are zeros. 

If a matrix is in row echelon form and each nonzero row has a leading entry of 1 and each leading 1 is the only nonzero entry in its column, then the matrix is said to be in reduced row echelon form. For example,

1   0   0   5   6
0   1   0   5   7
0   0   1   2   3

A pivot position in a matrix is a location that corresponds to a leading 1 in the reduced echelon form of the matrix. A pivot column is a column that contains a pivot position.

The Row Reduction Algorithm

Step 1
Begin with the leftmost nonzero. This is a pivot column. The pivot position is at the top.

Step 2
Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.

Step 3
Use row replacement operations to create zeros in all positions below the pivot.

Step 4
Repeat steps 1 - 3 to the submatrix that remains until there are no more nonzero rows to modify.

Step 5
Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, use a scaling operation to make it 1.

Linear Algebra (1)

Any system of linear equations can be transformed into a row equivalent system of equations or in other words one with the same solution set by using elementary row operations.

Elementary Row Operations
1.) Interchange two rows.
2.) Scale a row by a non-zero constant.
3.) Add a multiple of row to another row.

Consider the following matrix

1  0  1
0  1  1

Here, x = 1 and y = 1. If the rows are switched then the solution remains the same. If I scale a row, say row two by a constant c, then cy = c which means that cy/c = c/c but c/c = 1. So, y = 1; the solution remains the same. Now if I add row 1 to row 2 and replace row 1 with the result, then

1  1  2
0 1  1

Row 1 is now x + y = 2. So, this line has a y-intercept of 2 and a slope of -1 but still intersects row 2 at (1,1) because 1 + 1 = 2; a solution to the equation. In fact, no matter the multiple c of row 2 when added to row 1, which results in x + cy = c + 1 will always ensure that (1,1) is a solution since (1) + c(1) = c + 1.

Linear Algebra (0)

A linear equation is an equation that can be written in the form 

a1x1 + a2x2 + ... + anxn = b

where a1,a2,...,an and b are real or complex numbers. A linear system is a collection of one or more linear equations involving the same variables. A solution of a system is a list s1, s2, ..., sn of numbers that make each equation a true statement when the values of s1, s2, ... , sn are substituted for x1, x2, ... , xn, respectively. The set of all possible solutions is called the solution set of the linear system. Two linear systems are equivalent if they have the same solution set.

A system of linear equations has

1.) no solution, or

2.) exactly one solution, or

3.) infinitely many solutions.

A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.

Matrix Notation

The equations 

x    -   2y +  z  =  0

          2y -  8z =  8

-4x + 5y + 9z = -9

can be compactly recorded in the following ways. First, by using a coefficient matrix, and by using an augmented matrix.

  1   -2    1

  0    2   -8

-4     5    9

  1   -2    1    0
  0    2   -8    8
-4     5    9   -9