Section 21

Sum and intersection of subspaces

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Problem 21.1

Prove that the space of square matrices of order nn is the direct sum of the subspace of symmetric matrices and the subspace of skew-symmetric matrices of the same order.

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Problem 21.2

Prove that the space of polynomials of degree at most nn is the direct sum of the subspace of even polynomials of degree at most nn and the subspace of odd polynomials of degree at most nn.

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Problem 21.3
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(1)

Prove that an nn-dimensional vector space is the direct sum of the subspace of vectors all of whose coordinates are equal to each other, and the subspace of vectors the sum of whose coordinates equals 0.

(2)

A matrix AA of nn rows is given. Prove that the nn-dimensional arithmetic space is the direct sum of the linear span of the columns of AA and the subspace of solutions of the system of linear equations ATx=0A^{T} \mathbf{x} = \mathbf{0}.

Problem 21.4

Prove that the nn-dimensional arithmetic space is the direct sum of the vector subspaces spanned by the systems of vectors a1,…,ak\mathbf{a}_{1}, \ldots , \mathbf{a}_{k} and b1,…,bl\mathbf{b}_{1}, \ldots , \mathbf{b}_{l}:

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(1)

n=2,a1=c30,a2=c32,b1=c35n = 2, \quad \mathbf{a}_{1} = \mathbf{c}_{30}, \quad \mathbf{a}_{2} = \mathbf{c}_{32}, \quad \mathbf{b}_{1} = \mathbf{c}_{35}. c30=[−11],c32=[2−2],c35=[14]\mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{32} = \left[\begin{smallmatrix} 2 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{35} = \left[\begin{smallmatrix} 1 \\ 4 \end{smallmatrix}\right].

(2)

n=3,a1=c141,a2=c146,b1=c66,b2=c140n = 3, \quad \mathbf{a}_{1} = \mathbf{c}_{141}, \quad \mathbf{a}_{2} = \mathbf{c}_{146}, \quad \mathbf{b}_{1} = \mathbf{c}_{66}, \quad \mathbf{b}_{2} = \mathbf{c}_{140}. c141=[211],c146=[141],c66=[111],c140=[444]\mathbf{c}_{141} = \left[\begin{smallmatrix} 2 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{146} = \left[\begin{smallmatrix} 1 \\ 4 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{140} = \left[\begin{smallmatrix} 4 \\ 4 \\ 4 \end{smallmatrix}\right].

(3)

n=4,a1=c198,a2=c196,b1=c197,b2=c166,b3=c206n = 4, \mathbf{a}_{1} = \mathbf{c}_{198}, \mathbf{a}_{2} = \mathbf{c}_{196}, \mathbf{b}_{1} = \mathbf{c}_{197}, \mathbf{b}_{2} = \mathbf{c}_{166}, \mathbf{b}_{3} = \mathbf{c}_{206}. c198=[1113],c196=[1213],c197=[1122],c166=[1111],c206=[2233]\mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{196} = \left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{206} = \left[\begin{smallmatrix} 2 \\ 2 \\ 3 \\ 3 \end{smallmatrix}\right].

(4)

n=4,a1=c196,a2=c198,a3=c202,a4=c199,b1=c166n = 4, \mathbf{a}_{1} = \mathbf{c}_{196}, \mathbf{a}_{2} = \mathbf{c}_{198}, \mathbf{a}_{3} = \mathbf{c}_{202}, \mathbf{a}_{4} = \mathbf{c}_{199}, \mathbf{b}_{1} = \mathbf{c}_{166}, b2=c204\mathbf{b}_{2} = \mathbf{c}_{204}. c196=[1213],c198=[1113],c202=[1013],c199=[3−572],c166=[1111],c204=[3333]\mathbf{c}_{196} = \left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{199} = \left[\begin{smallmatrix} 3 \\ -5 \\ 7 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{204} = \left[\begin{smallmatrix} 3 \\ 3 \\ 3 \\ 3 \end{smallmatrix}\right].

Problem 21.5

Decompose the given vector x\mathbf{x} of the nn-dimensional arithmetic space into a sum of two vectors, one of which lies in P\mathcal{P}, and the other in Q\mathcal{Q}, where P\mathcal{P} is the linear span of the system of vectors a1,…,ak\mathbf{a}_{1}, \ldots , \mathbf{a}_{k}, and Q\mathcal{Q} is the linear span of the system of vectors b1,…,bl\mathbf{b}_{1}, \ldots , \mathbf{b}_{l}. Verify the uniqueness of the decomposition:

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(1)

n=2,x=c29,a1=c28,b1=c30n = 2, \quad \mathbf{x} = \mathbf{c}_{29}, \quad \mathbf{a}_{1} = \mathbf{c}_{28}, \quad \mathbf{b}_{1} = \mathbf{c}_{30}. c29=[32],c28=[−16],c30=[−11]\mathbf{c}_{29} = \left[\begin{smallmatrix} 3 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{28} = \left[\begin{smallmatrix} -1 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right].

(2)

n=3,x=c120,a1=c84,a2=c83,b1=c66n = 3, \quad \mathbf{x} = \mathbf{c}_{120}, \quad \mathbf{a}_{1} = \mathbf{c}_{84}, \quad \mathbf{a}_{2} = \mathbf{c}_{83}, \quad \mathbf{b}_{1} = \mathbf{c}_{66}. c120=[2−1−5],c84=[431],c83=[123],c66=[111]\mathbf{c}_{120} = \left[\begin{smallmatrix} 2 \\ -1 \\ -5 \end{smallmatrix}\right], \mathbf{c}_{84} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right].

(3)

n=3,x=c145,a1=c84,a2=c83,b1=c66n = 3, \quad \mathbf{x} = \mathbf{c}_{145}, \quad \mathbf{a}_{1} = \mathbf{c}_{84}, \quad \mathbf{a}_{2} = \mathbf{c}_{83}, \quad \mathbf{b}_{1} = \mathbf{c}_{66}. c145=[20−1],c84=[431],c83=[123],c66=[111]\mathbf{c}_{145} = \left[\begin{smallmatrix} 2 \\ 0 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{84} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right].

(4)

n=3,x=c139,a1=c84,a2=c83,b1=c66n = 3, \quad \mathbf{x} = \mathbf{c}_{139}, \quad \mathbf{a}_{1} = \mathbf{c}_{84}, \quad \mathbf{a}_{2} = \mathbf{c}_{83}, \quad \mathbf{b}_{1} = \mathbf{c}_{66}. c139=[−2−2−2],c84=[431],c83=[123],c66=[111]\mathbf{c}_{139} = \left[\begin{smallmatrix} -2 \\ -2 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{84} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right].

(5)

n=4,x=c200,a1=c166,a2=c198,a3=c207,b1=c202n = 4, \mathbf{x} = \mathbf{c}_{200}, \mathbf{a}_{1} = \mathbf{c}_{166}, \mathbf{a}_{2} = \mathbf{c}_{198}, \mathbf{a}_{3} = \mathbf{c}_{207}, \mathbf{b}_{1} = \mathbf{c}_{202}, b2=c205\mathbf{b}_{2} = \mathbf{c}_{205}. c200=[−18−65],c166=[1111],c198=[1113],c207=[1112],c202=[1013],c205=[0011]\mathbf{c}_{200} = \left[\begin{smallmatrix} -1 \\ 8 \\ -6 \\ 5 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{207} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{205} = \left[\begin{smallmatrix} 0 \\ 0 \\ 1 \\ 1 \end{smallmatrix}\right].

Problem 21.6

Find the projection of the given vector x\mathbf{x} of the nn-dimensional arithmetic space onto the vector subspace P\mathcal{P} parallel to the vector subspace Q\mathcal{Q}, where P\mathcal{P} is the linear span of the system of vectors a1,…,ak\mathbf{a}_{1}, \ldots , \mathbf{a}_{k}, and Q\mathcal{Q} is the linear span of the system of vectors b1,…,bl\mathbf{b}_{1}, \ldots , \mathbf{b}_{l}:

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(1)

n=2,x=c32,a1=c30,b1=c34n = 2, \quad \mathbf{x} = \mathbf{c}_{32}, \quad \mathbf{a}_{1} = \mathbf{c}_{30}, \quad \mathbf{b}_{1} = \mathbf{c}_{34}. c32=[2−2],c30=[−11],c34=[13]\mathbf{c}_{32} = \left[\begin{smallmatrix} 2 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{34} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right].

(2)

n=2,x=c37,a1=c30,b1=c34n = 2, \quad \mathbf{x} = \mathbf{c}_{37}, \quad \mathbf{a}_{1} = \mathbf{c}_{30}, \quad \mathbf{b}_{1} = \mathbf{c}_{34}. c37=[26],c30=[−11],c34=[13]\mathbf{c}_{37} = \left[\begin{smallmatrix} 2 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{34} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right].

(3)

n=2,x=c35,a1=c30,b1=c34n = 2, \quad \mathbf{x} = \mathbf{c}_{35}, \quad \mathbf{a}_{1} = \mathbf{c}_{30}, \quad \mathbf{b}_{1} = \mathbf{c}_{34}. c35=[14],c30=[−11],c34=[13]\mathbf{c}_{35} = \left[\begin{smallmatrix} 1 \\ 4 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{34} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right].

(4)

n=3,x=c146,a1=c66,a2=c121,a3=c122,b1=c145n = 3, \mathbf{x} = \mathbf{c}_{146}, \mathbf{a}_{1} = \mathbf{c}_{66}, \mathbf{a}_{2} = \mathbf{c}_{121}, \mathbf{a}_{3} = \mathbf{c}_{122}, \mathbf{b}_{1} = \mathbf{c}_{145}. c146=[141],c66=[111],c121=[−320],c122=[−231],c145=[20−1]\mathbf{c}_{146} = \left[\begin{smallmatrix} 1 \\ 4 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{121} = \left[\begin{smallmatrix} -3 \\ 2 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{122} = \left[\begin{smallmatrix} -2 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{145} = \left[\begin{smallmatrix} 2 \\ 0 \\ -1 \end{smallmatrix}\right].

(5)

n=4,x=c201,a1=c166,a2=c199,b1=c197,b2=c198n = 4, \mathbf{x} = \mathbf{c}_{201}, \mathbf{a}_{1} = \mathbf{c}_{166}, \mathbf{a}_{2} = \mathbf{c}_{199}, \mathbf{b}_{1} = \mathbf{c}_{197}, \mathbf{b}_{2} = \mathbf{c}_{198}. c201=[1−75−2],c166=[1111],c199=[3−572],c197=[1122],c198=[1113]\mathbf{c}_{201} = \left[\begin{smallmatrix} 1 \\ -7 \\ 5 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{199} = \left[\begin{smallmatrix} 3 \\ -5 \\ 7 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right].

Problem 21.7

Find the dimension and a basis of the sum and intersection of the vector subspaces of the nn-dimensional arithmetic space spanned by the systems of vectors a1,…,ak\mathbf{a}_{1}, \ldots , \mathbf{a}_{k} and b1,…,bl\mathbf{b}_{1}, \ldots , \mathbf{b}_{l}:

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(1)

n=2,a1=c34,a2=c37,a3=c35,b1=c30,b2=c36n = 2, \mathbf{a}_{1} = \mathbf{c}_{34}, \mathbf{a}_{2} = \mathbf{c}_{37}, \mathbf{a}_{3} = \mathbf{c}_{35}, \mathbf{b}_{1} = \mathbf{c}_{30}, \mathbf{b}_{2} = \mathbf{c}_{36}, b3=c32\mathbf{b}_{3} = \mathbf{c}_{32}. c34=[13],c37=[26],c35=[14],c30=[−11],c36=[3−3],c32=[2−2]\mathbf{c}_{34} = \left[\begin{smallmatrix} 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{37} = \left[\begin{smallmatrix} 2 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{35} = \left[\begin{smallmatrix} 1 \\ 4 \end{smallmatrix}\right], \mathbf{c}_{30} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{36} = \left[\begin{smallmatrix} 3 \\ -3 \end{smallmatrix}\right], \mathbf{c}_{32} = \left[\begin{smallmatrix} 2 \\ -2 \end{smallmatrix}\right].

(2)

n=3,a1=c116,a2=c121,a3=c119,b1=c122,b2=c125n = 3, \mathbf{a}_{1} = \mathbf{c}_{116}, \mathbf{a}_{2} = \mathbf{c}_{121}, \mathbf{a}_{3} = \mathbf{c}_{119}, \mathbf{b}_{1} = \mathbf{c}_{122}, \mathbf{b}_{2} = \mathbf{c}_{125}, b3=c138\mathbf{b}_{3} = \mathbf{c}_{138}. c116=[421],c121=[−320],c119=[−140],c122=[−231],c125=[5313],c138=[7012]\mathbf{c}_{116} = \left[\begin{smallmatrix} 4 \\ 2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{121} = \left[\begin{smallmatrix} -3 \\ 2 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{119} = \left[\begin{smallmatrix} -1 \\ 4 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{122} = \left[\begin{smallmatrix} -2 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{125} = \left[\begin{smallmatrix} 5 \\ 3 \\ 13 \end{smallmatrix}\right], \mathbf{c}_{138} = \left[\begin{smallmatrix} 7 \\ 0 \\ 12 \end{smallmatrix}\right].

(3)

n=3,a1=c66,a2=c139,a3=c140,b1=c94,b2=c141n = 3, \mathbf{a}_{1} = \mathbf{c}_{66}, \mathbf{a}_{2} = \mathbf{c}_{139}, \mathbf{a}_{3} = \mathbf{c}_{140}, \mathbf{b}_{1} = \mathbf{c}_{94}, \mathbf{b}_{2} = \mathbf{c}_{141}, b3=c117\mathbf{b}_{3} = \mathbf{c}_{117}. c66=[111],c139=[−2−2−2],c140=[444],c94=[321],c141=[211],c117=[532]\mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{139} = \left[\begin{smallmatrix} -2 \\ -2 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{140} = \left[\begin{smallmatrix} 4 \\ 4 \\ 4 \end{smallmatrix}\right], \mathbf{c}_{94} = \left[\begin{smallmatrix} 3 \\ 2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{141} = \left[\begin{smallmatrix} 2 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{117} = \left[\begin{smallmatrix} 5 \\ 3 \\ 2 \end{smallmatrix}\right].

(4)

(р) n=3,a1=c83,a2=c142,a3=c143,b1=c84,b2=c144,b3=c117n = 3, \mathbf{a}_{1} = \mathbf{c}_{83}, \mathbf{a}_{2} = \mathbf{c}_{142}, \mathbf{a}_{3} = \mathbf{c}_{143}, \mathbf{b}_{1} = \mathbf{c}_{84}, \mathbf{b}_{2} = \mathbf{c}_{144}, \mathbf{b}_{3} = \mathbf{c}_{117}. c83=[123],c142=[011],c143=[112],c84=[431],c144=[110],c117=[532]\mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{142} = \left[\begin{smallmatrix} 0 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{143} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{84} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{144} = \left[\begin{smallmatrix} 1 \\ 1 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{117} = \left[\begin{smallmatrix} 5 \\ 3 \\ 2 \end{smallmatrix}\right].

(5)

n=3,a1=c66,a2=c116,a3=c145,b1=c122,b2=c146n = 3, \mathbf{a}_{1} = \mathbf{c}_{66}, \mathbf{a}_{2} = \mathbf{c}_{116}, \mathbf{a}_{3} = \mathbf{c}_{145}, \mathbf{b}_{1} = \mathbf{c}_{122}, \mathbf{b}_{2} = \mathbf{c}_{146}, b3=c147\mathbf{b}_{3} = \mathbf{c}_{147}. c66=[111],c116=[421],c145=[20−1],c122=[−231],c146=[141],c147=[5−2−1]\mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{116} = \left[\begin{smallmatrix} 4 \\ 2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{145} = \left[\begin{smallmatrix} 2 \\ 0 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{122} = \left[\begin{smallmatrix} -2 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{146} = \left[\begin{smallmatrix} 1 \\ 4 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{147} = \left[\begin{smallmatrix} 5 \\ -2 \\ -1 \end{smallmatrix}\right].

(6)

n=3,a1=c83,a2=c84,a3=c120,b1=c66,b2=c121n = 3, \mathbf{a}_{1} = \mathbf{c}_{83}, \mathbf{a}_{2} = \mathbf{c}_{84}, \mathbf{a}_{3} = \mathbf{c}_{120}, \mathbf{b}_{1} = \mathbf{c}_{66}, \mathbf{b}_{2} = \mathbf{c}_{121}, b3=c122\mathbf{b}_{3} = \mathbf{c}_{122}. c83=[123],c84=[431],c120=[2−1−5],c66=[111],c121=[−320],c122=[−231]\mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{84} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{120} = \left[\begin{smallmatrix} 2 \\ -1 \\ -5 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{121} = \left[\begin{smallmatrix} -3 \\ 2 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{122} = \left[\begin{smallmatrix} -2 \\ 3 \\ 1 \end{smallmatrix}\right].

(7)

n=4,a1=c196,a2=c200,a3=c217,b1=c211,b2=c218n = 4, \mathbf{a}_{1} = \mathbf{c}_{196}, \mathbf{a}_{2} = \mathbf{c}_{200}, \mathbf{a}_{3} = \mathbf{c}_{217}, \mathbf{b}_{1} = \mathbf{c}_{211}, \mathbf{b}_{2} = \mathbf{c}_{218}, b3=c219\mathbf{b}_{3} = \mathbf{c}_{219}. c196=[1213],c200=[−18−65],c217=[010−58],c211=[14−15],c218=[3−263],c219=[4258]\mathbf{c}_{196} = \left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{200} = \left[\begin{smallmatrix} -1 \\ 8 \\ -6 \\ 5 \end{smallmatrix}\right], \mathbf{c}_{217} = \left[\begin{smallmatrix} 0 \\ 10 \\ -5 \\ 8 \end{smallmatrix}\right], \mathbf{c}_{211} = \left[\begin{smallmatrix} 1 \\ 4 \\ -1 \\ 5 \end{smallmatrix}\right], \mathbf{c}_{218} = \left[\begin{smallmatrix} 3 \\ -2 \\ 6 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{219} = \left[\begin{smallmatrix} 4 \\ 2 \\ 5 \\ 8 \end{smallmatrix}\right].

(8)

n=4,a1=c196,a2=c198,a3=c202,b1=c166,b2=c204n = 4, \mathbf{a}_{1} = \mathbf{c}_{196}, \mathbf{a}_{2} = \mathbf{c}_{198}, \mathbf{a}_{3} = \mathbf{c}_{202}, \mathbf{b}_{1} = \mathbf{c}_{166}, \mathbf{b}_{2} = \mathbf{c}_{204}, b3=c197\mathbf{b}_{3} = \mathbf{c}_{197}. c196=[1213],c198=[1113],c202=[1013],c166=[1111],c204=[3333],c197=[1122]\mathbf{c}_{196} = \left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{204} = \left[\begin{smallmatrix} 3 \\ 3 \\ 3 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right].

(9)

n=4,a1=c166,a2=c196,a3=c197,b1=c203,b2=c205n = 4, \mathbf{a}_{1} = \mathbf{c}_{166}, \mathbf{a}_{2} = \mathbf{c}_{196}, \mathbf{a}_{3} = \mathbf{c}_{197}, \mathbf{b}_{1} = \mathbf{c}_{203}, \mathbf{b}_{2} = \mathbf{c}_{205}, b3=c206\mathbf{b}_{3} = \mathbf{c}_{206}. c166=[1111],c196=[1213],c197=[1122],c203=[2222],c205=[0011],c206=[2233]\mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{196} = \left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{203} = \left[\begin{smallmatrix} 2 \\ 2 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{205} = \left[\begin{smallmatrix} 0 \\ 0 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{206} = \left[\begin{smallmatrix} 2 \\ 2 \\ 3 \\ 3 \end{smallmatrix}\right].

(10)

n=4,a1=c166,a2=c198,a3=c196,a4=c202,b1=c207,b2=c204,b3=c197,b4=c205n = 4, \mathbf{a}_{1} = \mathbf{c}_{166}, \mathbf{a}_{2} = \mathbf{c}_{198}, \mathbf{a}_{3} = \mathbf{c}_{196}, \mathbf{a}_{4} = \mathbf{c}_{202}, \mathbf{b}_{1} = \mathbf{c}_{207}, \mathbf{b}_{2} = \mathbf{c}_{204}, \mathbf{b}_{3} = \mathbf{c}_{197}, \mathbf{b}_{4} = \mathbf{c}_{205}. c166=[1111],c198=[1113],c196=[1213],c202=[1013],c207=[1112],c204=[3333],c197=[1122],c205=[0011]\mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{198} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{196} = \left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{207} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{204} = \left[\begin{smallmatrix} 3 \\ 3 \\ 3 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{205} = \left[\begin{smallmatrix} 0 \\ 0 \\ 1 \\ 1 \end{smallmatrix}\right].

(11)

(р) n=4,a1=[1−110],a2=[1100]n = 4, \quad \mathbf{a}_{1} =\left[\begin{smallmatrix} 1 \\ - 1 \\ 1 \\ 0\end{smallmatrix}\right], \quad \mathbf{a}_{2} =\left[\begin{smallmatrix} 1 \\ 1 \\ 0 \\ 0\end{smallmatrix}\right], a3=[2012],a4=[2011],b1=[4614]\mathbf{a}_{3} =\left[\begin{smallmatrix} 2 \\ 0 \\ 1 \\ 2\end{smallmatrix}\right], \quad \mathbf{a}_{4} =\left[\begin{smallmatrix} 2 \\ 0 \\ 1 \\ 1\end{smallmatrix}\right], \quad \mathbf{b}_{1} =\left[\begin{smallmatrix} 4 \\ 6 \\ 1 \\ 4\end{smallmatrix}\right], b2=[1120],b3=[11−23],b4=[2411]\mathbf{b}_{2} =\left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 0\end{smallmatrix}\right], \quad \mathbf{b}_{3} =\left[\begin{smallmatrix} 1 \\ 1 \\ - 2 \\ 3\end{smallmatrix}\right], \quad \mathbf{b}_{4} =\left[\begin{smallmatrix} 2 \\ 4 \\ 1 \\ 1\end{smallmatrix}\right].

Problem 21.8

Find the dimension and a basis of the sum and intersection of the vector subspaces of the space of square matrices of order 3, spanned by the systems of matrices A202,A201,A209,A204\mathbf{A}_{202}, \mathbf{A}_{201}, \mathbf{A}_{209}, \mathbf{A}_{204} and A256,A205,A257,A258\mathbf{A}_{256}, \mathbf{A}_{205}, \mathbf{A}_{257}, \mathbf{A}_{258}. A202=[12221−22−21],A201=[21−131−2101],A209=[1−3−1−27232−4],A204=[−231362121],A256=[001026164],A205=[123234345],A257=[0544−42029],A258=[0534−6−4−1−45]\mathbf{A}_{202} = \left[\begin{smallmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{smallmatrix}\right], \mathbf{A}_{201} = \left[\begin{smallmatrix} 2 & 1 & -1 \\ 3 & 1 & -2 \\ 1 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{209} = \left[\begin{smallmatrix} 1 & -3 & -1 \\ -2 & 7 & 2 \\ 3 & 2 & -4 \end{smallmatrix}\right], \mathbf{A}_{204} = \left[\begin{smallmatrix} -2 & 3 & 1 \\ 3 & 6 & 2 \\ 1 & 2 & 1 \end{smallmatrix}\right], \mathbf{A}_{256} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 2 & 6 \\ 1 & 6 & 4 \end{smallmatrix}\right], \mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right], \mathbf{A}_{257} = \left[\begin{smallmatrix} 0 & 5 & 4 \\ 4 & -4 & 2 \\ 0 & 2 & 9 \end{smallmatrix}\right], \mathbf{A}_{258} = \left[\begin{smallmatrix} 0 & 5 & 3 \\ 4 & -6 & -4 \\ -1 & -4 & 5 \end{smallmatrix}\right].

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Problem 21.9

Find the dimension and a basis of the sum and intersection of the vector subspaces of the space of polynomials of degree at most 3, spanned by the systems of polynomials 1+2t+t31 + 2t + t^{3}, 1+t+t2,t−t2+t31 + t + t^{2}, t - t^{2} + t^{3} and 1+t2,1+3t+t3,3t−t2+t31 + t^{2}, 1 + 3t + t^{3}, 3t - t^{2} + t^{3}.

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Problem 21.10

Using the notion of the sum of two vector subspaces, prove the inequality rg⁡(A+B)⩽rg⁡A+rg⁡B\operatorname {rg}(A + B) \leqslant \operatorname {rg} A + \operatorname {rg} B, where AA and BB are two matrices of the same size m×nm \times n.

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Problem 21.11

Prove that the sum L\mathcal{L} of two vector subspaces P\mathcal{P} and Q\mathcal{Q} is a direct sum if and only if at least one vector x∈Lx\in \mathcal{L} is uniquely represented in the form x=y+zx = y + z, where y∈P,z∈Qy\in \mathcal{P}, z\in \mathcal{Q}.

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Problem 21.12

Let P\mathcal{P} and Q\mathcal{Q} be two vector subspaces of a finite-dimensional vector space. Prove that:

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(1)

if the sum of the dimensions of P\mathcal{P} and Q\mathcal{Q} is greater than the dimension of the whole space, then the intersection P∩Q\mathcal{P} \cap \mathcal{Q} contains a nonzero vector

(2)

if the dimension of the sum of P\mathcal{P} and Q\mathcal{Q} is one more than the dimension of their intersection, then one of these subspaces is contained in the other.

Problem 21.13

Prove that for any vector subspace P\mathcal{P} of a finite-dimensional vector space there exists another subspace Q\mathcal{Q} such that the whole space is the direct sum of P\mathcal{P} and Q\mathcal{Q}.

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Problem 21.14

Let L,M,N\mathcal{L}, \mathcal{M}, \mathcal{N} be three vector subspaces; P=(L∩N)+(M∩N),Q=(L+M)∩N\mathcal{P} = (\mathcal{L} \cap \mathcal{N}) + (\mathcal{M} \cap \mathcal{N}), \mathcal{Q} = (\mathcal{L} + \mathcal{M}) \cap \mathcal{N}

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(1)

Prove that P⊆Q\mathcal{P} \subseteq \mathcal{Q}.

(2)

Is the case P≠Q\mathcal{P} \neq \mathcal{Q} possible? Give a corresponding example.

Problem 21.15

Prove that the sum of the vector subspaces L(1),…,L(k)\mathcal{L}^{(1)}, \ldots , \mathcal{L}^{(k)} coincides with the set of vectors representable in the form x=x1+…+xkx = x_{1} + \ldots + x_{k}, where xi∈L(i),i=1,…,kx_{i} \in \mathcal{L}^{(i)}, i = 1, \ldots , k.

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Problem 21.16

Let L,M,N\mathcal{L}, \mathcal{M}, \mathcal{N} be three vector subspaces. Prove that

L+M+N=(L+M)+N=L+(M+N) \mathcal{L} + \mathcal{M} + \mathcal{N} = (\mathcal{L} + \mathcal{M}) + \mathcal{N} = \mathcal{L} + (\mathcal{M} + \mathcal{N})
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Problem 21.17

Prove that the sum L\mathcal{L} of the vector subspaces L(1),…,L(k)\mathcal{L}^{(1)}, \ldots , \mathcal{L}^{(k)} is a direct sum if and only if at least one vector x∈Lx\in \mathcal{L} is uniquely represented in the form x=x1+…+xkx = x_{1} + \ldots + x_{k}, where xi∈L(i)x_{i} \in \mathcal{L}^{(i)} (a generalization of problem 21. 11).

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