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Thread: matrix problems

  1. #1
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    matrix problems

    Let A= (1/3 1/3 1/3)^T. Think of A as an operator from R^1(a one roll matrix with real number) to R^3(a matrix with 3 rows and one column) via matrix vector multiplication.

    My questions are that how can I show that

    1) the operator is one to one
    2) how can i find 2 linear left- inverses for A
    3) how can i find a left inverse for A that is not linear

    A= (1/3 1/3 1/3)^T this, T mean transpose

    Plz help
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  2. #2
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    Quote Originally Posted by oxxiissiixxo View Post
    Let A= (1/3 1/3 1/3)^T. Think of A as an operator from R^1(a one roll matrix with real number) to R^3(a matrix with 3 rows and one column) via matrix vector multiplication.

    My questions are that how can I show that

    1) the operator is one to one
    2) how can i find 2 linear left- inverses for A
    3) how can i find a left inverse for A that is not linear

    A= (1/3 1/3 1/3)^T this, T mean transpose

    Plz help
    the operator corresponding to $\displaystyle A$ is $\displaystyle T: \mathbb{R} \longrightarrow \mathbb{R}^3$ defined by $\displaystyle T(x)=\begin{bmatrix}\frac{x}{3} & \frac{x}{3} & \frac{x}{3} \end{bmatrix}^T.$ it's obviously 1-1 because $\displaystyle T(x)=\bold{0}$ iff $\displaystyle x=0.$ now define the maps: $\displaystyle S_1,S_2: \mathbb{R}^3 \longrightarrow \mathbb{R}$ by:

    $\displaystyle S_1( \begin{bmatrix}x & y & z \end{bmatrix}^T)=3x, \ S_2( \begin{bmatrix}x & y & z \end{bmatrix}^T)=3y.$ clearly $\displaystyle S_1,S_2$ are linear and $\displaystyle S_1T(x)=S_2T(x)=x.$ hence $\displaystyle S_1,S_2$ are left inverse of $\displaystyle T.$ also, for example, the map $\displaystyle S_3: \mathbb{R}^3 \longrightarrow \mathbb{R}$

    defined by: $\displaystyle S( \begin{bmatrix}x & y & z \end{bmatrix}^T)=x(x-y+3),$ is a non-linear map which is a left inverse of $\displaystyle T$ because $\displaystyle S_3T(x)=x.$
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  3. #3
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    $\displaystyle

    $
    $\displaystyle S_1( \begin{bmatrix}x & y & z \end{bmatrix}^T)=3x, \ S_2( \begin{bmatrix}x & y & z \end{bmatrix}^T)=3y.
    $


    how do you get this. do you just manually make the S1 and S1 = to 3x and 3 y?
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  4. #4
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    $\displaystyle S( \begin{bmatrix}x & y & z \end{bmatrix}^T)=x(x-y+3),$ is a non-linear map which is a left inverse of $\displaystyle T$ because $\displaystyle S_3T(x)=x.$[/quote]

    how come this S_3of T(x) is x for this?
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