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Thread: 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 $\displaystyle B_2$ coordinates of a point that is given to you in the $\displaystyle B_1$ coordinate system, so the point itself does not change.

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

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

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

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

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

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

    Thus,

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

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

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

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