Assign4 - Math 136 Assignment#4 1 Let A be an m n matrix...

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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 ) = Ax + 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 - 4 0 1 2 3 - 2 3 0 5 and b = - 1 3 - 1 4 . Is b in the range of the linear transformation x Ax ? 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) T ( x 1 , x 2 , x 3 , x 4 ) = (0 , x 1 + x 2 , x 3 - 3 x 4 , 2 x 1 + x 3 + x 4 ). (b) T ( x 1 , x 2 , x 3 , x 4 ) = ( x 2 1 , x 2 , x 3 + x 4 , - x 2 + x 4 ).
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