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Homework Help: Linear Algebra and matrix

  1. Jan 23, 2006 #1
    a few questions
    a) can a 3x4 matrix have independant columns? rows? Explain
    if i were to reduce to row echelon form then i could potentially have 4 leading 1s. I m not quite sure about this.
    if i were to reduce this 3 x 4 matrix into row echelon form then the number of rows is less than the number of variables. SO the answer is no.

    b) if A is a 4 x3 matrix and rank A = 2, can A have independant columns? rows? Explain
    ok rank A means that out of the 4 rows only 2 are non zero when A is in row echelon form. Potentially 3 leading 1s in the columns so at least 2 of the columns may be dependant on each other. So independant columns are not possible.
    Indepednat rows not possible.

    c) Can a non square matrix has its rows indepedant and its columns independant?
    im not sure about this. If A (MxN) then for m rows A has n unknowns so it is not possible to have indepdnatn rows. As for the columns i ahve no idea.

    If A is m x n and B is n x m show taht AB = 0 iff [itex] col B \subseteq null A [/itex]
    suppose AB = 0
    let columns of B = [itex]C_{i}[/itex]
    rows of A = [itex]R_{i}[/itex]
    for all i
    then [tex] R_{i} C_{i} = 0 [/itex] if Ci = 0 for all i. Thus Ci belongs to null A
    Suppose [tex] col B \subseteq null A [/tex]
    then anything times a column of B is zero. Thus AB = 0
    Is this proof adequate?

    your input is greatly appreciated!
     
  2. jcsd
  3. Jan 24, 2006 #2

    TD

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    Homework Helper

    Remember that a rank of a matrix is equal to the ranks of its transpose, this allows you to intechange rows & columns for your explanation.

    A 3x4 matrix can have at most a rank 3, so what does that tell you about the maximum number of linearly indepedant rows/columns?
    For the second, the rank is now given - what does this tell you again?
    You can use the same argument again for a non-square m x n. Suppose m > n, then the maximal possible ranks is n.
     
  4. Jan 24, 2006 #3
    for a 3x4 matrix\
    the rank can be at most 3
    that means it can have at most 3 linearly independant rows
    4 linear independant columns

    for hte second
    for rank A = 2
    then there are 2 indpendant rows
    so at most only 2 indpendant columns?
     
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