Linear independence

Linear independence

Lessons

We say that a set of vectors {v1,,vpv_1, \cdots , v_p} in Rn\Bbb{R}^n is linearly independent if:
v1x1+v2x2++vpxp=0 v_1 x_1+v_2 x_2+\cdots+v_p x_p=0

gives only the trivial solution. In other words, the only solution is:

solution to a set of linearly independent vectors


We say that a set of vectors {v1,,vpv_1, \cdots , v_p} in Rn\Bbb{R}^n is linearly dependent if:
v1x1+v2x2++vpxp=0 v_1 x_1+v_2 x_2+\cdots+v_p x_p=0


gives a non-trivial solution. In other words, they are linearly dependent if it has a general solution (aka has free variable).

We can determine if the vectors is linearly independent by combining all the columns in matrix (denoted as A) and solving for

Ax=0 Ax=0

Fast way to tell if 2 or more vector are linearly dependent
1. The vectors are multiples of one another
2. There are more vectors than there are entries in each vector.
3. There is a zero vector
  • 1.
    Linear Independence Overview:
    a)
    Linearly independent
    • Definition of linear independence
    • Trivial solution

    b)
    Linearly dependent
    • Definition of linear dependence
    • Non-trivial solutions

    c)
    Fast ways to determine linear dependence
    • Vectors are multiples of one another
    • # of Vectors > # of Entries in each vector
    • Zero Matrix


  • 2.
    Determining Linear independence by solving
    Determine if the following vectors are linearly independent:
    is this vector linearly independent

  • 3.
    Determine if the matrix is linearly independent by solving Ax=0Ax=0:
    linearly independent matrix

  • 4.
    Determining Linear dependence by inspection
    Determine by inspection if the following vectors are linearly dependent:
    a)
    determine if vector is linearly dependent by inspection

    b)
    determine if vector is linearly dependent by inspection

    c)
    determine if vector is linearly dependent by inspection

    d)
    determine if vector is linearly dependent by inspection


  • 5.
    Linear dependence/independence with unknown constant
    Find the value(s) of kk for which the vectors are linearly dependent.
    find the unknown value if the vectors are linearly dependent