By Robert McDowell; Tornheim, Leonard Thrall
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1, -1, 2, 1), (0, 1, 1, 0), (2, O; 0, 0). What is the kernel of T? 8. If T is a linear transformation that sends respectively, into the kernel of T. (1, 1, 1), (3, 2, -1), (5, 4, 1), (1, 0, 0), What (1, 0, 0), (0, 1, 0), (0, 0, 1), find the matrix for T, and find What is the dimension of the range of T? 9. If T has the matrix 2 A find Jf"(T). 1 1 4 3 0 1 2 3 What is the dimension of 1""3T? 4. Multiplication of linear transformations. Z. For any vector IX in r, fJ = IXT is a vector in if'". Z; Let three vector spaces over a field§', let T be a linear transformation of furthermore y is uniquely determined by IX.
We recall that two linear transformations S and T are equal if they define the same mapping; that is, if they have the same domain rand if for each vector oc in f' we have ocS= oc T. Suppose that T is a linear transformation off' into if'" and that subspace off'. Then we can define a mapping Sof equation yS = yT for all yin tFI. tFt is a tFI into if'" by the Clearly Sis a linear transformation. LINEAR TRANSFORMATIONS AND MATRICES 34 Except for the trivial case "Y 'Pt, the transformations S and = equal since they do not have the same domain.
Q) for which (a) What is the matrix for Twhen (4, -7, 8). Q) into = linear (1, 0, 2). Q)? which Find bases relative to which the '-.. ___ 7. Find the matrix of the linear transformation T that sends (0, 1, 0), (0, 0, l),respectively,into is the image of (2, -1, 1) under T? (1, -1, 2, 1), (0, 1, 1, 0), (2, O; 0, 0). What is the kernel of T? 8. If T is a linear transformation that sends respectively, into the kernel of T. (1, 1, 1), (3, 2, -1), (5, 4, 1), (1, 0, 0), What (1, 0, 0), (0, 1, 0), (0, 0, 1), find the matrix for T, and find What is the dimension of the range of T?