2.3 Composition of Trans. 1

# 2.3 Composition of Trans. 1 - 2.3 2.3 Composition of Linear...

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2.3. 2.3. Composition of Linear Transformations and Matrix Multiplication 1. (a) F ([ UT ] γ α = [ U ] γ β [ T ] β α ) (p.88 Theorem 2.11) (b) T (p.91 Theorem 2.14) (c) F ([ U ( w )] γ = [ U ] γ β [ w ] β for all w W ) (d) T (Since I V ( v j ) = v j , 1 i,j n [I V ] α =I n (e) F In case T:V V, [ T 2 ] α = [ T · T ] α = [ T ] α [ T ] α = ([ T ] α ) 2 (f) F If I 6 =A= ± 1 2 1 2 3 2 - 1 2 , then A 2 =I (cf) A= ± 0 1 1 0 (g) F (T: F n F m T= L A for some A M m × n ( F ) (h) F If A= ± 0 0 1 0 , then A 2 = 0 even though A 6 = 0 The cancelation property for multiplication in ﬁelds is not valid for matrices. (i) T (p.93 Theorem 2.15(c)) (j) T (p.89 Deﬁnition) 2. 87 PNU-MATH

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2.3. (a) (1) A(2B+3C)= ± 20 - 9 18 5 10 8 (2) (AB)D= ( 29 - 26 ) (3) A(BD)= ( 29 - 26 ) (b) (1) A t = ± 2 - 3 4 5 1 2 (2) A t B= ± 23 19 0 26 - 1 10 (3) BC t = 12 16 19 (4) CB= ( 27 7 9 ) (5) CA= ( 20 26 ) 3. (a) (1) [ U ] γ β U (1) = U (1 + 0 · x + 0 · x 2 ) = (1 , 0 , 1) U ( x ) = U (0 + 1 · x + 0 · x 2 ) = (1 , 0 , - 1) U ( x 2 ) = U (0 + 0 · x + 1 · x 2 ) = (0 , 1 , 0) [ U ] γ β = 1 1 0 0 0 1 1 - 1 0 (2) [ T ] β T (1) = T (0 · (3 + x ) + 2 · 1) = 2 = 2 · 1 + 0 · x + 0 · x 2 88 PNU-MATH
2.3. T

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2.3 Composition of Trans. 1 - 2.3 2.3 Composition of Linear...

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