Vectors: True/False and Linear Equations

• True or false questions about vectors

Question 1

• The value of 𝑣⋅(𝑣×𝑤) is always 0, true
• 𝑣×𝑤 is orthogonal to both 𝑣 and 𝑤
• 𝑣⋅(𝑣×𝑤) = 0

Question 2

• It is never true that 𝑣×𝑤=𝑤×𝑣, false
• 𝑣×𝑤=𝑤×𝑣 if 𝑣×𝑤=0

Question 3

• Given a non-zero vector 𝑣 in 3-space, there is a non-zero vector 𝑤 such that 𝑣×𝑤=0, true
• If 𝑤 is a scalar multiple of 𝑣 then 𝑣×𝑤=𝑣×𝑠𝑣 = 𝑠(𝑣×𝑣) = 0

Topic 1: Linear Algebra

• Solving a system of linear equations
• Linear systems arise in many applications in engineering, including electrical networks, mechanical frameworks, traffic flow, economic models, curve and surface fitting, and optimisation problems
• Concepts in vector geometry and linear algebra.
• Linear systems can be understood from both vector geometry and linear algebra

1. 1 Systems of linear equations

• A linear equation is an equation in which each term is either a constant or a constant multiplied by a variable to the first power.
• Unknowns x1, x2, ..., xn can be written in the form a1x1 + a2x2 + ... + anxn = b
• A linear equation does not involve any products or roots of variables.
• All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or exponential functions.
• Example linear equations: 2x + 3y = 0, y = -x + 4z + 6, x1 - 2x2 - 3x3 + x4 = 7
• Example non-linear equations: y=x^2, x + 3(sqrt y) = 5, 3x + 2y - xz = 0, y = sin(x)
• A system of linear equations (or linear system), is a set or group of linear equations which must all be true at the same time

Example 1: solving a circuit problem

• Solving a circuit problem uses Kirchhoff's laws

Example 2: curve fitting

• A quadratic polynomial graph passes through the points (1,16), (2,6), and (3,58)
• General form of the quadratic polynomial in x is p(x) = ax^2 + bx + c where a, b and c are constants
• Since p(1) = 16, we substitute x=1 into the formula p(x) = ax^2 + bx + c, this gives the equation a + b + c = 16
• Two more equations are obtained similarly from x=2 and x=3:
  • x = 1, a + b + c = 16
  • x = 2, 4a + 2b + c = 6
  • x = 3, 9a + 3b + c = 58
• If this system of three linear equations in the three unknowns a, b, c, finds the parabola that passes through the above points:
  • p(2) = 6, a(2)^2 + b(2) + c = 6, 4a + 2b + c = 6
  • p(3) = 58, a(3)^2 + b(3) + c = 58, 9a + 3b + c = 58

Comments

• Real problems can give rise to much larger systems that involve thousands of equations and unknowns
• Systematic method for solving is vital; the method can be programmed on a computer
• Example 2 had the same number of equations as the number of unknowns
• Other problems may lead to systems where the number of equations are greater than the number of unknowns (Example 1) or less than the number of unknowns
• Questions to ask:
  • Can always find a solution to the system?
  • Is it possible for the system to have no solutions?
  • Could there be more than one solution?

Exercise 2 A simple case - one linear equation in one variable

• Solving equations:
  • 3x = 2, x = 2/3, one unique solution
  • 0x = 2, no solutions, inconsistent
  • 0x = 0, any value of x is a solution (x in R), infinitely many solutions
• A solution to an equation in one variable: A value of x that satisfies the equation
• The solution, or the general solution, to an equation: All the values of x that satisfy the equation
• Cases for the solution of ax = b:
  • One unique solution when a != 0. Then x = b/a.
  • Infinitely many solutions when a=0 and b=0.
  • No solutions when a=0 and b!=0

Exercise 3 One linear equation in two variables

• Describing the solution set of the linear equation 3x + 5y = 15:
  • 2 variables but only 1 equation (constraint) must be satisfied
  • Any value can be chosen for one of the variables, but then this value determines the value of the other variable
• Has infinitely many solutions
• One of the variables can be any real number (called a free variable)
• To describe this, use a parameter
  • Set x=t, t in R
• Equation becomes 3t + 5y = 15, 5y = 15 - 3t, y = 3 - (3/5)t
• The solution set in vector form is (x,y) = (t, 3 - (3/5)t)
• Or, l = (0,3) + t(1, -(3/5)), t in R, vector equation for a line

Can a single linear equation in two variables have just one solution?

• ax + by = c cannot have just one solution
• If it is consistent it will always have infinitely many solutions
• All the points on the line defined by the equation
• If we have 2 variables, we need at least 2 equations to get one solution Think about possible values of a, b, and c in ax + by = c
• Yes! If a = 0 and b = 0 and c != 0
• eg, 0x + 0y = 1.

Exercise 4 2x2 systems two equation in two variables

• Graphing straight lines

What do we mean by the solution, or the general solution, to a 2 x 2 system of equations?

• All the ordered pairs (x,y) that satisfy both equations

What are the three cases for the solution of a 2x2 system of equations?

• One solution, the two lines intersecting at a point
• No solution, the two lines are inconsistent
• Infinitely many solutions, the equations represent the same line

1. 2 Systems of equation in tree or more variables

• What does the equation x + y + z = 2 represent geometrically?
• x + y + z = 2 represent a plane in 3-space
• This system has 3 variables and 1 equation (constraint)
• If the system is consistent, there will be 2 free variables
• We assign parameters to these free variables to write the solution in vector form
• Let y=s and z=t, s, t in R
• Equation becomes x + s + t = 2, x = 2-s-t,
• Solution is (x,y,z) = (2-s-t, s, t) = (2-s-t, 0+s+0t, 0+0s+t)
• l = (2,0,0) + s(-1,1,0) + t(-1,0,1) a vector parametric equation for a plane

Quick review quiz

• Which equations are linear?
  • Equations with number
• What are the three possible types of solutions to a linear system with the same number of equations and variables?
  • No solutioms, 1 solution, infinite solutions

3 x 3 systems

• Geometrically, a linear system of three equations in three variables can be thought of as three planes in 3-space

• What are the three possible cases for the solutions of a 3 x 3 system of equations? Think of this in terms of the relationship of three planes in 3-space

  • One solution, where three planes intersect at a point
  • No solution, for example when all three planes are parallel There are other ways three planes can give rise to the corresponding system of equations having no solution?
  • Infinitely many solutions, for example, when the planes intersect in a line.

• An m x n linear system

  • A solution to an m x n system of equations, is an ordered n-tuple of numbers (x1, x2, ..., xn) that satisfies all the equations of the system
  • Aim to find all solutions to a given system: the general solution or solution set

• If a system of equations has no solution, it is an inconsistent system

Triangular systems and back-substitution

• Simplest equation to solve is of the form ax = b
• Simplest systems to solve are diagonal systems
• Example solution 5x = 15, 2y=4, 3z = -6
• Example upper triangular system:
  • 4x + y + 2z = 15, 2y + z = 4, 3z = -6
• Start with the 3rd equation as it has just one unknown.
• Substitute this value back into the 2nd equation and solving for y gives y = 3.
• Substitute the values for z and y into the 1st equation to get x = 4

Exercise 6 back-substitution

• Solve the following upper triangular linear system to find what it is

Echelon Form

• A system of linear equations is in Echelon Form (EF) if the first variable in each equation is further to the right as we move down the equations
• The pivot variables (or leading variables) of a linear system in EF are those variables that are first in each of the equations
• The variables that are not pivot variables are free variables
• If each equation in the echelon form of the system contains a pivot variable, then the system is consistent and can be solved by back-substitution

1. 4 A systematic approach: elimination and back-substitution

• Algorithm used must be readily extendible to problems with large systems

• The solution to the new transformed system has to be the same as the solution to the original system

• Two linear systems are equivalent if they have the same solution set

• Three types of elementary operations can be performed on a system of equations to get an equivalent system

• These elementary operations can be undone (they are reversible or “invertible”), and they preserve the original solutions

  • Multiplying an equation through by a nonzero constant
  • Swapping two equations
  • Adding a multiple of one equation to another equation

• Two linear systems are equivalent if they have the same solution set

• If two systems are related by an elementary operation then they are equivalent

• Any consistent system of equations can be put in echelon form by a sequence of elementary operations

• Follow a systematic elimination process by first using elementary operations to get the system into echelon form and then solving this equivalent system by back-substitution

Gaussian elimination

• Name of mathematician Carl Friedrich Gauss because he described it in a paper detailing the computations he made in order to determine the orbit of the asteroid Pallas

Exercise 8 system of linear equations

• Use the first equation to eliminate x from the second equation, by multiplying the first equation by 3 and subtracting it from the second equation to get a new second equation

Augmented Matrix

• Augmented matrices shorthand notation for complicated systems
• Consisting of coefficient matrix and the constant terms
• The matrix is called the coefficient matrix for the system of equations
• A matrix is just a rectangular array of numbers
  • "Matrices" is the plural of "matrix”: one matrix, two matrices

Exercise 9 Augmented matrices

• Each column of the coefficient matrix corresponds to one variable

Elementary Row Functions

• Starting with an augmented matrix, elementary row operations is used to obtain a row equivalent matrix
• Operations correspond to the elementary operations for systems of equations since each row from the augmented matrix is one equation from the system
• Operations: These preserve original solution
  • Multiplying a row through by a nonzero constant.
  • Interchanging two rows.
  • Adding a multiple of one row to another different row.

Gaussian Elimination After Row Reduction

• The matrix is in row echelon form (REF)
• The first non-zero element in any row is called a pivot and the corresponding variable is called a pivot variable (all entries below the pivots are zero
• The corresponding system of linear equations is in echelon form

Comments

• A matrix is in row echelon form if all non-zero rows (rows with at least one non-zero element) are above all zero rows
• If each pivot (the first non-zero number from the left in a row) is always strictly to the right of the pivots above it
• Putting the system in augmented matrix form and using row operations to reduce the matrix to row-echelon form is called "Gaussian elimination" or "row reduction"
• Can continue using elementary row operations to reduce a matrix further into reduced row echelon form (RREF)
  • All pivots are scaled to 1; this makes them “leading 1s”
  • All entries above a leading 1 are also zero
• Back substitution is trivial (that is, easily done) and the solutions can be written down
• This extension is called "Gauss-Jordan elimination"
• In practice these extras steps are not used when solving linear systems since no efficiency is gained
• Some textbooks require that pivots should be scaled to 1 (easily done with a row operation) in their definition of row echelon form
• Can safely ignore this extra requirement though sometimes such scaling makes the arithmetic easier as it avoids complicated fractions appearing in the augmented matrix.

Exercise 13 Gaussian elimination

Solve the following systems of equations by Gaussian elimination If a row states 0x1+ 0x2 + 0x3 = 1, the system of equations is inconsistent (has no solution)

Flow

• Flow in equals the flow out

Last Equations

• In row-reduced matrices, the last equation indicates the nature of the solution

Homogenous Linear Systems

• Consider a linear system of equations written with all the variable terms on the left-hand side of each equation and the constant term on the right-hand side
• If all the right-hand side terms of the equations are 0 then the system is called homogeneous
• Homogeneous linear systems always have solutions
• The type of the solution space is determined by the number of variables and the number of pivots