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Math Help - Linear Transformations

  1. #1
    Junior Member
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    Linear Transformations

    Verify that the mapping T : R4 -> R2 defined by
    T (x1; x2; x3; x4) = (x3 - x2; 5x1 + 3x4)
    is a linear transformation.

    Find a basis for Im (T ) and Ker (T ).
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  2. #2
    Senior Member Dinkydoe's Avatar
    Joined
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    Showing linearity:

    Choose 2 vectors in v_1,v_2\in \mathbb{R}^4.

    v_1 = (a_1,a_2,a_3,a_4)
    v_2 = (b_1,b_2,b_3,b_4)
    Choose \lambda_1,\lambda_2\in \mathbb{R}

    We want to show that T(\lambda_1v_1+\lambda_2v_2)= \lambda_1T(v_1)+\lambda_2T(v_2):

    Can you do that? It's just writing out.

    Observe that v\in ker (T) \Leftrightarrow T(v)= (x_3-x_2,5x_1+3x_4) = (0, 0). Hence v satisfies x_3= x_2, x_4 = -\frac{5}{3}x_1.

    Find two independant vectors in v_1,v_2\in \mathbb{R}^4 that satisfy these relations and you have a basis for ker(T).
    Find 2 more independant vectors: v_3,v_4\in \mathbb{R}^4 and you have a basis for Im(T)
    (Dim(Im(T))+ Dim(ker(T)) = 4).
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