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Linear independence if there is a column of zeros

  1. Feb 7, 2012 #1
    1. The problem statement, all variables and given/known data
    If, in a matrix, there is a column of all zeros, does this mean the given vector/matrix is linearly dependent?
    An example would be:
    [1 2 0 4]
    [2 3 0 1]
    [5 2 0 7]

    A few questions to clear up some possible misconceptions:
    1) The matrix above is a 4-dimensional vector?
    2) The given vector has 4 unknowns, but a column of zeros, therefore the system is linearly dependent.

    Thanks for your help!!!
  2. jcsd
  3. Feb 7, 2012 #2
    I don't think the matrix you have is a vector, vectors are either row vectors (1xm matrices) or column vectors (nx1 matrices). For this system of linear equations, each row represents a different equation. The system is then linearly independent if each row is independent, ie. row reduction doesn't yield a row of zeros. A column of zeros doesn't affect this linear independence. This is what it affects: if you take a 4x1 column vector, say [a,b,c,d]^T, and multiply it on the left by your matrix, it maps this vector to a 3x1 vector based on the column vectors of your matrix. The zero-column (3rd column) of your matrix maps the third row of your [a,b,c,d]^T to 0 in all rows of its image, which ends up being [a+2b+4d,2a+3b+d,5a+2b+7d]^T.
  4. Feb 7, 2012 #3


    Staff: Mentor

    A set of vectors can be linearly dependent or linearly independent, but it doesn't usually make any sense to describe a single vector as being dependent or independent.
    No. The matrix on its own belongs to a vector space of dimension 12. The rows of the matrix are vectors in R4, a 4-dimensional vector space. The columns of the matrix are vectors in R3, a 3-dimensional vector space.
    What given vector? Does the matrix above represent a system of three equations in four unknowns?

    You need to be more specific about what you're asking before we can give meaningful answers.
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