canonical form

Examples of canonical form in the following topics:

• Simplifying Matrices With Row Operations

  • Using elementary operations, Gaussian elimination reduces matrices to row echelon form.
  • Upper triangle form: A square matrix is called upper triangular if all the entries below the main diagonal are zero.
  • Use elementary row operations on the augmented matrix $[A|b]$ to transform $A$ to upper triangle form.
  • It is now in reduced row echelon form, or row canonical form.  
  • Use elementary row operations to put a matrix in simplified form

• Further Simplification of Matrices

  • Matrices containing zeros below each pivot are said to be in row echelon form.
  • In diagonal form, we remove any zeros from the diagonal and add them below and above.
  • Use elementary row operations on matrix [A|b] to transform A into diagonal form.
  • Then, use elementary row operations to transform A into diagonal form:
  • A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination.

• Application of Systems of Inequalities: Linear Programming

  • You may have noticed that we had been given inequalities, such as $3a+2b+c \leq 10$, but standard form calls for equalities, or equations.
  • Standard form also requires the objective function to be a minimization.
  • A linear program in standard form can be represented as a tableau of the form
  • If the columns of A can be rearranged so that it contains the p-by-p identity matrix (the number of rows in A), then the tableau is said to be in canonical form.

• Graphing Quadratic Equations in Vertex Form

  • Quadratic equations may take various forms.
  • You have already seen the standard form:
  • Another common form is called vertex form, because when a quadratic is written in this form, it is very easy to tell where its vertex is located.
  • The vertex form is given by:
  • It is more difficult to convert from standard form to vertex form.

• Converting the Conic Equation of a Parabola to Standard Form

• Linear Equations in Standard Form

  • Another way of arranging a linear equation is in standard form.
  • In the standard form, a linear equation is written as:
  • For example, consider an equation in slope-intercept form: $y = -12x +5$.
  • However, the zero of the equation is not immediately obvious when the linear equation is in this form.
  • Convert linear equations to standard form and explain why it is useful to do so

• Slope-Intercept Equations

  • One of the most common representations for a line is with the slope-intercept form.
  • Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and $y$-intercept.  
  • Let's write an equation in slope-intercept form with $m=-\frac{2}{3}$, and $b=3$.  
  • If an equation is not in slope-intercept form, solve for $y$ and rewrite the equation.
  • Finally, divide all terms by $-6$ to get the slope-intercept form $y=2x-1$.  

• Standard Form and Completing the Square

  • The standard form of a quadratic equation is useful for completing the square, which is used to graph the equation.
  • Completing the square is a technique for converting a quadratic polynomial of the form:
  • The expression inside the parenthesis is of the form (x − constant).
  • This form of quadratic equation is known as the "standard form" for graphing parabolas in algebra; from this equation, it is simple to determine the x-intercepts (y = 0) of the parabola, a process known as "solving" the quadratic equation.

• What is a Quadratic Function?

  • Quadratic equations are second order polynomials, and have the form $f(x)=ax^2+bx+c$.
  • A quadratic function is of the general form:
  • Quadratic functions can be expressed in many different forms.
  • The form written above is called standard form.
  • is known as factored form, where $x_1$ and $x_2$ are the zeros, or roots, of the equation.

• Point-Slope Equations

  • Write an equation of a line in Point-Slope Form (given a point and a slope)  Convert to Slope-Intercept Form
  • Write an equation of a line in Point-Slope Form (given two points)  Convert to Slope-Intercept Form
  • Converting to slope-intercept form means to solve the equation for $y$.
  • The only difference is the form that they are written in.
  • Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation