3 Read questions carefully, and answer all questions as they are stated. 4 Prese

Question

3 Read questions carefully, and answer all questions as they are stated. 4 Present your solutions in a elear and well-organized manner. Aluvags show all work. Cirele the correct answer whenever applicable Have a photo ID and a sessional student card on your desk ready. 1. (6pts) Circle the correct answer, then prove or disprove the following statements (a)Let a € R2 be a fixed noazero vector aud let T,:R, R, be the functiou defined by T.(r) = YES NO r+a. The function T, is a linear transformation (b) The transformation T,4:R3R, induced by A-1-2-64|sone-to-one? YES NO (e) For every linear transformation T:R-R"we have T(0) = 0. YES NO

Explanation / Answer

(a).The statement is False. Ta(x+y) = (x+y)+a, while Ta(x)+Ta(y) = (x+a)+(y+a )= (x+y) +2a Ta(x+y). This means that Ta does not preserve vector addition. Hence Ta is not a linear transformation.

(b). The statement is False. The RREF of A is

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This implies that the columns of A are not linearly independent. Hence T is not one-to-one.

( c). The statement is True. Since T is a linear transformation, it preserves scalar multiplication so that T(0) = T(0.x) = 0.T(x) = 0.

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