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Linear Algebra - Basis of column space

  1. Mar 9, 2014 #1
    1. The problem statement, all variables and given/known data
    Let A be the matrix
    A =
    1 −3 −1 2
    0 1 −4 1
    1 −4 5 1
    2 −5 −6 5

    (a) Find basis of the column space. Find the coordinates of the dependent columns relative
    to this basis.
    (b) What is the rank of A?
    (c) Use the calculations in part (a) to find a basis for the row space.


    2. Relevant equations
    ---

    3. The attempt at a solution

    I used Gauss-Jordan operations on the matrix to solve it down to :

    1 0 -13 5
    0 1 -4 1
    0 0 1 0
    0 0 0 0

    From here we can see which columns are linearly independent and which are dependent. But I don't understand what they want me to write for a solution for the coordinates.

    Would they simply be:

    (1,0,-13)
    (0,1,-4)
    (0,0,1)

    ??

    b) Not sure exactly what this means even after researching, how do I determine the rank ?

    c) I feel I can do after I complete a)

    Thanks
     
  2. jcsd
  3. Mar 9, 2014 #2

    Zondrina

    User Avatar
    Homework Helper

    For part 1, by definition, ##Col(A) = span\{a_1, ... a_n\}## where ##a_1, ... a_n## are the linearly independent columns of ##A##.

    The basis happens to be the set ##\{a_1, ... a_n\}## (without the "span" portion).

    Also, do you know about coordinate vectors?
     
  4. Mar 9, 2014 #3
    Your given matrix has 4 numbers in each column. That is each column is in R^4. So how can the span of {(1,0,-13), (0,1,-4), (0,0,1)} be subset of R^4????. You need to get your definitions done perfectly before you can solve these problems.
     
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