1.3 Subspaces - 1.3 1.3 Subspaces 1(a F(p.1 Definition of...

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Unformatted text preview: 1.3. 1.3. Subspaces 1 . (a) F (p.1 Definition of subspace) (b) F (0 / ∈ ∅ ) (c) T ( V and { ∅ } are subspaces of V ) (d) F (p.19 Theorem 1.4) (e) F (f) F (p.18 Example 4) (g) F ((0 , , 0) ∈ W , but (0 , , 0) / ∈ R 2 ) 2 . (b), (c), (e), (f), (g) are not square matrices (a) -5, (d) 12, (h) -6 3 . ∀ A,B ∈ M m × n ( F ) ,a,b ∈ F (1 ≤ i ≤ m, 1 ≤ j ≤ n ) ( aA + bB ) t ij = ( aA + bB ) ji = ( aA ) ji + ( bB ) ji = a ( A ) ji + b ( B ) ji = aA t ij + bB t ij = ( aA t + bB t ) ij ∴ ( aA + bB ) t = aA t + bB t 4 . ( A t ) t ij = ( A t ) ji = A ij 5 . ( A + A t ) t = A t + ( A t ) t = A t + A = A + A t ∴ A + A t is symmetric 9 PNU-MATH 1.3. 6 . tr ( aA + bB ) = n ∑ i =1 ( aA + bB ) ii = n ∑ i =1 ( aA ) ii + n ∑ i =1 ( bB ) ii = atr ( A ) + btr ( B ) 7 . A = a 11 a 22 O O a 33 a 44 ⇒ A t = A ∴ A is symmetric 8 . (a) Yes (b) No ((0 , , 0) / ∈ W 2 ) (c) Yes (d) Yes (e) No ((0 , , 0) / ∈ W 5 ) (f) No x + y / ∈ W 6 ) , ∀ x,y ∈ W 6 9 . (1) W 1 ∩ W 3 = { } is a subspace of R 3 (2) W 1 ∩ W 4 = W 1 is a subspace of R 3 (3) W 3 ∩ W 4 = { ( a 1 ,a 2 ,a 3 ) ∈ R 3 | 3 a 1 = 11 a 2 , 3 a 3 = 23 a 2 } is a subspace of R 3 10 . (i) W 1 is a subspace of F n (ii) W 2 is not a subspace of F n ( ∵ (0 , , 0) / ∈ W 2 ) 10 PNU-MATH 1.3. 11 . No (The given set is not closed under addition) (Example) ∀ f,g ∈ W Let f ( x ) = a n x n + a n- 1 x n- 1 + ··· + a , degf = n g ( x ) = b n x n + b n- 1 x n- 1 + ··· + b , degg = n If b n =- a n , then deg ( f + g ) = n- 1 ∴ f + g / ∈ W ∴ W is not a subspace of V 15 . Yes 17 . ( ⇒ ) Theorem 1.3 ( ⇐ ) W is a subspace of...
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1.3 Subspaces - 1.3 1.3 Subspaces 1(a F(p.1 Definition of...

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