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MAT 272 Assignment 3 - solutions

MAT 272 Assignment 3 - solutions - The same can be shown...

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MAT 272 Assignment #3 Due: October 6, 2009 1) Plot the four vertices of the unit square ((0,0), (1,0), (0,1), (1,1)) and also plot the images of the four vertices using the following transformations T( x ) = A x on . Describe the effect of each one. a. A= Reflect in y-axis b. A= Reflect in x-axis c. A= Reflect in y=x d. A= Rotation in origin. 2) Suppose S and T are both linear transformations from to . Show that the compositions S(T( x )) and T(S( x )) are also linear transformations. Recall S( x+y ) = S( x )+S( y ) and T( x+y ) = T( x )+T( y ) by definition. Also S(a x ) = aS( x ) and T(a x ) = aT( x ) by definition. Property 1) S(T( u+v )) = S(T( u )+ T( v )) = S(T( u ))+S(T( v ). Property 2) S(T(c u )) = S(cT( u )) = cS(T( u )) This shows that S(T) is a linear transformation,

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Unformatted text preview: The same can be shown for T(S). 3) Suppose . Solve for a , b , c and d. This leads to the system a-b=8, b+c=1, 3d+c=7 2a-4d=6. In matrix form this is which reduces to So a = 5, b = -3, c = 4 and d = 1. 4) Write out the 4×4 matrix A=[ a ij ] that satisfies a ij = i j-1 . 5) Find the inverse of the matrix where 0 ≤ 2 . 6) Let , , , a = 4 and b = -7. Verify the following facts: a. (AB)C = A(BC) (AB)C= = = A(BC)= = = b. (B + C)A = BA + CA (B + C)A = = = BA + CA = = = c. ( a C) T = a (C T ) ( a C) T = = = a (C T ) = = = d. a ( b C) = ( ab )C a ( b C) = = = ( ab )C = = = e. (AB) T = B T A T (AB) T = = = B T A T = = =...
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MAT 272 Assignment 3 - solutions - The same can be shown...

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