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Thread: proving row space column space

  1. #1
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    proving row space column space

    A , B are nXn matrices
    and
    AB=(A)^t
    t-is transpose
    prove that the space spanned by A's row equals the space spanned by A's columns
    i know that there dimentions are equals
    so in order to prove equality i need to prove that one is a part of the other
    how to do it?
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  2. #2
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    Re: proving row space column space

    Quote Originally Posted by transgalactic View Post
    A , B are nXn matrices
    and
    AB=(A)^t
    t-is transpose
    prove that the space spanned by A's row equals the space spanned by A's columns
    i know that there dimentions are equals
    so in order to prove equality i need to prove that one is a part of the other
    how to do it?
    an element in the row space of $\displaystyle A$ is in the form $\displaystyle A^tx$, where $\displaystyle x$ in a vector. an element in the column space of $\displaystyle A$ is in the form $\displaystyle Ay$, where $\displaystyle y$ is a vector. so if the vector $\displaystyle x$ is given and we let $\displaystyle Bx = y,$ then $\displaystyle A^tx = Ay$. this proves that the row space of $\displaystyle A$ is a subspace of the column space of $\displaystyle A$.
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  3. #3
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    Re: proving row space column space

    each column i of (AB)_i=A*B_i
    i was told by my proff that that column i of AB is a member from the span of the columns of A

    but i dont get this result
    suppose the member of B_i column is (c1,c2,..,cn)
    so the multiplication of A by the B_i column
    we get then the first member is dot product from the first row with (c1,c2,..,cn)
    i cant see how its a variation from the A columns?
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  4. #4
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    Re: proving row space column space

    the key is this simple fact that if $\displaystyle x$ is a vector with entries $\displaystyle x_1, \ldots, x_n$ and $\displaystyle v_1, \ldots v_n$ are the columns of an $\displaystyle n \times n$ matrix $\displaystyle C$, then $\displaystyle Cx = x_1v_1 + \ldots + x_n v_n$, which is an element of the column space of $\displaystyle C$. it should be clear now that $\displaystyle C^tx$ is an element of the row space of $\displaystyle C$.
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  5. #5
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    Re: proving row space column space

    i cant get the diagram
    of a coefficient next to each of A's columns from this coulmn by matrix multiplication
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  6. #6
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    Re: proving row space column space

    Quote Originally Posted by transgalactic View Post
    i cant get the diagram
    of a coefficient next to each of A's columns from this coulmn by matrix multiplication
    well, i suggest you take $\displaystyle 2 \times 2$ matrices first to get an idea.
    let $\displaystyle C = \begin{pmatrix}a & b \\ c & d \end{pmatrix}.$ so the columns are $\displaystyle v_1=\begin{pmatrix}a \\ c \end{pmatrix}$ and $\displaystyle v_2 = \begin{pmatrix}b \\ d \end{pmatrix}$. now let $\displaystyle x = \begin{pmatrix}x_1 \\ x_2 \end{pmatrix}$. show that $\displaystyle Cx=x_1v_1 + x_2v_2.$
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