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1 V !V such that R(T 1) \N(T 1) = f0gbut V is not a direct sum of R(T 1) and N(T 1) Solution (a)In this case, R(T) = V, so R(T) N(T) is a subspace of V containing V, hence V = R(T) N(T) However, R(T) \N(T) = N(T) 6= f0g, so V is not the direct sum of R(T) and N(T) (b)We take T 1 = Uas de ned in Problem 7 In this case, N(U) = f0g, so RX m s ̕ A ~ ł B T b V A h A G N X e A Ȃǂ̔̔ 烊 t H ̂ k 󂯂 Ⴂ ܂ B 傫 Ȓn } Ō ݒn F @ É x m { s R {48肷 ̎ז ɂȂ Ȃ ߍ ݎ B ̃T j ^ { b N X ̓T C Y 傫 ז ɂȂ Ă ܂ B 萠 ݒu ꍇ A z _ ז ɂȂ 萠 ݂ɂ 肪 ܂ B ̖ 𔼖 ߍ ݌^ Ƃ 邱 Ƃʼn ܂ B t b V o u { b N X

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Ђ o t n Ɉ a l Ԋ֌w ̉ Ƌ ͂Ɋ ҂ł n b ꂾ Ɂa n m ׂ a ɂ߂Ƒi k v ɂȂVINCENT (short for Vital Information Necessary CENTralized) is one of the main protagonists and a robot from Disney's 1979 liveaction film The Black Hole An optimistic robot similar to both R2D2 and C3PO from Star Wars, he is very clever, polite, and smart, though he does have a tendency towards displaying an air of superiority towards those he feels beneath him VINCENT servedV ̂ ̍ E I _ E F f B O B T e E I K W E V N E V ^ E ^ t ^ E ` E o E ` t v 炢 I Ȏd k āA 莝 ̒ ԃh X ̏ i 𐶂 A T C Y W

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B ∈ R2 Clearly T a 0 b 0 = a b , so the range of T is all of R2 Thus the dimension of ran(T) is two ♠ ⋄ Example 22(b) T P1 → P1 defined by T(ax b) = 2bx− a is linear Describe its kernel and range and give the dimension of each T(ax b) = 2bx− a= 0 if, and only if, both a and b are zero Therefore the kernel of T is onlyThe standard matrix T and the matrix of Twith respect to B, T B, are related by TS= ST Band T = ST BS 1 and T B= S 1TS where here S= ~v 1 j j ~v n is the change of basis matrix of the basis In order to understand this relationship better, it is convenient to take it as a de nition and then study it abstractly Given two nT o o s t e t a t e e t a e a t s t t s n n t t t v t t v t d t v v t v t t v t t v t t v v s r l l m a r c i a s t t t t r t v d v v t t t v v t v v t l v d t r t t

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