Solution sets of linear systems

Solution sets of linear systems

Lessons

A system of linear equations is homogeneous if we can write the matrix equation in the form Ax=0Ax=0.

A system of linear equations is nonhomogeneous if we can write the matrix equation in the form Ax=bAx=b.

We can express solution sets of linear systems in parametric vector form.

Here are the types of solutions a homogeneous system can have in parametric vector form:
1. With 1 free variable: x=tvx=tv
2. With 2 free variables: x=su+tvx=su+tv
3. With n free variables: x=av1+bv2++nvnx=av_1+bv_2+\cdots+nv_n

Here are the types of solutions a nonhomogeneous system can have in a parametric vector form:
1. With 1 free variable: x=p+tvx=p+tv
2. With 2 free variables: x=p+su+tvx=p+su+tv
3. With n free variables: x=p+av1+bv2++nvnx=p+av_1+bv_2+\cdots+nv_n
  • Introduction
    Solution Set of Linear Systems Overview:
    a)
    Homogeneous Systems
    Ax=0 Ax=0
    • Trivial, non-Trivial, and general solutions

    b)
    Solution Sets of Homogeneous Systems
    • Parametric Vector Equation with 1 free variable: x=tvx=tv
    • Parametric Vector Equation with 2 free variable x=su+tvx=su+tv
    • Parametric Vector Equation with more than 2 free variables

    c)
    NonHomogeneous Systems
    Ax=bAx=b
    • General Solutions with an extra vector

    d)
    Solution Sets of Nonhomogeneous Systems
    • Parametric Vector Equation with 1 free variable: : x=p+tvx=p+tv
    • Parametric Vector Equation with 2 free variables: x=p+su+tvx=p+su+tv
    • Parametric Vector Equation with more than 2 free variables

    e)
    Difference Between Homogeneous and Nonhomogeneous
    • Extra p=p= Translation


  • 1.
    Solution Sets of Homogeneous Systems
    Find the solution set of the homogeneous system in parametric vector form:
    x1+3x2+3x3=0 x_1+3x_2+3x_3=0
    x13x23x3=0 -x_1-3x_2-3x_3=0
    2x2+2x3=0 2x_2+2x_3=0

  • 2.
    Find the solution set Ax=0Ax=0 in parametric vector form if:
    find solution set in parametric vector form

  • 3.
    Solution Sets of Non-homogeneous Systems
    Find the solution set of the nonhomogeneous system in parametric vector form:
    x1+2x23x3=2 x_1+2x_2-3x_3=2
    2x1+x23x3=4 2x_1+x_2-3x_3=4
    x1+x2=2 -x_1+x_2=-2

  • 4.
    Comparing Homogeneous and Nonhomogeneous Systems
    Describe and compare the solution sets of 2x1+3x25x3=02x_1+3x_2-5x_3=0 and 2x1+3x25x3=42x_1+3x_2-5x_3=4

  • 5.
    Describe and compare the solution sets of x14x2+2x3=0x_1-4x_2+2x_3=0 and x14x2+2x3=3x_1-4x_2+2x_3=-3

  • 6.
    Parametric equation of a line and Translation
    Find the parametric equation of the line through Find the parametric equation of the line 1 parallel to Find the parametric equation of the line 2. Draw all the vectors and the line to show you obtained the line itself geometrically.