1. linear system and matrix algebra

• What is the general form of a linear equation in n variables?


  a1x1 + a2x2 + ··· + anxn = b

  
• What do the terms a1 to an represent in a linear equation?

  They are called the coefficients corresponding to the variables.

  
• In the equation x + 3y = 7, what are the variables?

  x and y

  
• What does the equation y = x – 0.5z + 4.5 represent?

  It is a linear equation in three variables x, y, and z, representing a plane in the xyz space.

  
• What is the standard form of a linear equation?

  All variables on the left and the constant term on the right.

  
• What is a key feature of a linear equation?

  Different variables can only be combined using addition or subtraction.

  
• What is an example of a non-linear equation?

  xy = 2

  
• Why is the equation sin(θ) + cos(φ) = 0.2 considered non-linear?


  Because it involves sine and cosine functions applied to the variables.
• What does the equation x12 + x22 + ··· + xn2 = 1 represent?

  It is non-linear because it involves the square of each variable.

  
• What is the geometrical representation of a linear equation with two variables?

  It represents a line in the xy-plane.

  
• What does a linear equation with three variables represent?

  It represents a flat plane in the xyz-space.

  
• What does it mean when a linear equation has four variables?

  It requires a 4-dimensional space to hold the three-dimensional object, which is beyond visualization.

  
• What is a system of linear equations?

  It is formed when a few linear equations are put together with the same set of variables in common.

  
• How are coefficients and constant terms denoted in a linear system?

  They are denoted by the letters a's and b's respectively.
• What is a particular solution of a linear system?

  It is a set of numbers that satisfy all the equations simultaneously.

  
• What is an example of a particular solution for a linear system?


  x1 = 1, x2 = 2, x3 = –1

  
• What does it mean if a linear system has no solution?

  It is called an inconsistent system due to contradictions among the equations.

  
• How can we determine if a linear system is inconsistent?

  By observing contradictions among the equations.
• What are the three possible cases for the number of solutions in a linear system?

  No solution, exactly one solution, or infinitely many solutions.
• What is a homogeneous linear system?


  A system where all constant terms are zero.
• What is the trivial solution of a homogeneous system?

  x1 = 0, x2 = 0, …, xn = 0

  
• Is a homogeneous system always consistent?

  Yes, a homogeneous system is always consistent regardless of the coefficients.
• What are the characteristics of linear equations?


  • Represent straight lines or flat planes.
  • Combine variables only through addition or subtraction.
  • Represent objects that are one dimension smaller than the space they occupy.

  
• What are the implications of having more than three variables in a linear equation?

  • Requires higher-dimensional space for representation.
  • Difficult to visualize geometrically.
  • Still applicable in mathematical modeling.

  
• What are the types of solutions for linear systems?

  • Inconsistent system: no solution.
  • Consistent system: at least one solution.
  • Infinitely many solutions: occurs when there are more than one solution.

  
• What is the difference between homogeneous and non-homogeneous systems?

  • Homogeneous: all constant terms are zero.
  • Non-homogeneous: at least one constant term is not zero.
• What is the significance of the trivial solution in homogeneous systems?

  • It is always a solution.
  • Indicates consistency of the system.
  • Can exist alongside other solutions.

  
• What is a homogeneous system of linear equations?

  A system where all constant terms are zero.

  
• How does a homogeneous system differ from a non-homogeneous system?

  A homogeneous system has all constant terms equal to zero, while a non-homogeneous system has at least one non-zero constant term.
• What is the trivial solution of a homogeneous system?

  It is when all variables are equal to zero, i.e., \(x_1 = 0, x_2 = 0, \ldots, x_n = 0\).

  
• Why is a homogeneous system always consistent?

  Because it always has at least the trivial solution.
• What does it imply if a homogeneous system has non-trivial solutions?

  It implies that the system has infinitely many solutions.
• What are the two properties regarding the number of solutions of homogeneous systems?

  1. A homogeneous system has either only the trivial solution or infinitely many solutions in addition to the trivial solution.
  2. A homogeneous system with more variables than equations has infinitely many solutions.

  
• How do two planes represented by a homogeneous system intersect in xyz-space?


  They intersect at least at the origin, which corresponds to the trivial solution.
• What are the two possible relative positions of two planes in a homogeneous system?

  They can either intersect along a line or overlap completely.
• What is an augmented matrix?

  It is a rectangular array that represents a linear system, including coefficients and constant terms.

  
• What does the augmented matrix of a linear system consist of?

  It consists of m horizontal rows for m equations and n+1 vertical columns for n variables and constant terms.

  
• What are the three types of elementary row operations on an augmented matrix?

  • Multiplying a row by a nonzero constant.
  • Interchanging two rows.
  • Adding a multiple of a row to another row.

  
• What happens when you multiply a row of an augmented matrix by a nonzero constant?

  All entries in that row are multiplied by the constant.

  
• What does interchanging two rows in an augmented matrix do?

  It swaps the positions of those two rows.