Math 136 Assignment #4 1. Let A be an m × n matrix and ~ b ∈ R m . Let T : R n → R m be the transformation deﬁned by T ( ~x ) = A~x + ~ b . Show that T is a linear transformation if and only if ~ b = ~ 0. (The transformation of the above form is called an “aﬃne” transformation.) 2. Let A = 1 3 9 2 1 0 3-40 1 2 3-2 3 0 5 and ~ b = -1 3-1 4 . Is ~ b in the range of the linear transformation ~x 7→ A~x ? Explain your answer. 3. Let T : R n → R m be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation T ( ~x ) = ~ 0 has a nontrivial solution. 4. Determine whether the following transformations are linear or not. If a transformation is linear, then ﬁnd the corresponding standard matrix and check that the transformation is onto or one-to-one. (a)
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This note was uploaded on 07/01/2008 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.