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Math Help - Linear Transformations

  1. #1
    Member kjchauhan's Avatar
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    Linear Transformations

    Please solve the Example:

    Determine the matrix [T: B_1, B_2] for the given L.T. T and the basis B_1 and B_2.

    B_1={(1,1),(1,0)} and B_2={(2,3),(4,5)}.

    I need Solution..
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  2. #2
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    One thing to keep in mind here is that you're just trying to find the B_2 coordinates of a point that is given to you in the B_1 coordinate system, so the point itself does not change.

    A point in the B_1 coordinate system has the coordinates (x, y) expressed as the linear combination of the B_1 basis vectors:

    x\begin{bmatrix}1 \\ 1\end{bmatrix} + y\begin{bmatrix}1 \\ 0\end{bmatrix}

    In the B_2 coordinate system, a point (u, v) will have coordinates of the expressed as linear combinations of the B_2 basis vectors:

    u\begin{bmatrix}2 \\ 3\end{bmatrix} + v\begin{bmatrix}4 \\ 5\end{bmatrix}

    These are the same point (we are just changing the coordinate system), so:

    x\begin{bmatrix}1 \\ 1\end{bmatrix} + y\begin{bmatrix}1 \\ 0\end{bmatrix} = u\begin{bmatrix}2 \\ 3\end{bmatrix} + v\begin{bmatrix}4 \\ 5\end{bmatrix}

    Thus,

    B_1\begin{bmatrix}x \\ y\end{bmatrix} = B_2\begin{bmatrix}u \\ v\end{bmatrix}, where B_1 = \begin{bmatrix}1 & 1\\ 1 & 0\end{bmatrix} and B_2 = \begin{bmatrix}2 & 4\\ 3 & 5\end{bmatrix}

    In order to get \begin{bmatrix}u \\ v\end{bmatrix} by itself, multiply both sides of the equation on the left by B_2^{-1} to get:

    \begin{bmatrix}u \\ v\end{bmatrix} = B_2^{-1} B_1\begin{bmatrix}x \\ y\end{bmatrix}

    So, the matrix that changes base from B_1 to B_2 is B_2^{-1} B_1.
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