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Math Help - Linear transformations and their inverse

  1. #1
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    Linear transformations and their inverse

    A couple questions:

    1)Consider the transformation T from R^2 to R^3 given by T [x1, x2] = x1[1, 2, 3] + x2[4, 5, 6]. Is this transformation linear? If so, find its matrix. Note: these matrices are vertical, I just didn't know how to type them vertically.

    2)Find an n x m matrix A such that Ax = 3x, for all x in R^n. Note: x is a vector, I just didn't know how to put a vector arrow above it.

    Thanks
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  2. #2
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    Quote Originally Posted by noles2188 View Post
    A couple questions:

    1)Consider the transformation T from R^2 to R^3 given by T [x1, x2] = x1[1, 2, 3] + x2[4, 5, 6]. Is this transformation linear? If so, find its matrix. Note: these matrices are vertical, I just didn't know how to type them vertically.
    You show that transformation is linear, of course, by showing that the definition of "linear transformation" holds:
    Is T\left(a\begin{bmatrix}x_1 \\ x_2\end{bmatrix}+b\begin{bmatrix}x_3 \\ x_4\end{bmatrix}\right)= aT\left(\begin{bmatrix}x_1 \\ x_2\end{bmatrix}\right)+ bT\left(\begin{bmatrix}x_3\\x_4\end{bmatrix}\right  )?

    To write it as a matrix, find T\left(\begin{bmatrix} 1 \\ 0\end{bmatrix}\right) and T\left(\begin{bmatrix} 0 \\ 1\end{bmatrix}\right). That will give you the two columns of the matrix.

    2)Find an n x m matrix A such that Ax = 3x, for all x in R^n. Note: x is a vector, I just didn't know how to put a vector arrow above it.
    This is easy! Think of it as A\vec{x}= 3 I\vec{x} where I is the identity matrix.

    Thanks
    To see the LaTex code I used for these, click on each formula.
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