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It satisfies Conditions (i), (ii’), (iii), (iv). 3) is the n-dimensional Euclidean space En . e. n x2j |x| = 1 2 f or x ∈ En . j=1 The number |x|, where x ∈ En , can be interpreted in two manners, either as the length of a vector or the distance of the point x from zero. Hence the number |x − y| denotes either the lenght of a vector which is a difference of vectors x and y or the distance of points x and y. 1. 4) x, y = xj yj = 0 (x, y ∈ En ). j=1 A vector x ∈ En is said to be normed if |x| = 1.

6. Mappings preserving all linear manifolds are called affine transformations. Describe all affine transformations in the spaces R2 and R3 . 7. ,p . ,n Prove that (A + B)T = AT + B T , (tA)T = tAT f or t ∈ F, (AT )T = A, (CA)T = AT C T (cf. 23)). 8. A square matrix A is said to be symmetric if AT = A and antisymmetric if AT = −A. Prove that every square matrix A is a sum of a symmetric matrix A+ and an antisymmetric A− and that this sum is uniquely determined, namely A = A+ + A− , where A+ = 1 (A + AT ), 2 A− = 1 (A − AT ).

An n-dimensional pre-Hilbert space over the field R is said to be an n-dimensional Euclidean space and it is denoted by En . A norm of an element x belonging to a pre-Hilbert space X is the function x = x, x . Observe that, by definition, x ≥ 0. By the distance of two points belonging to a pre-Hilbert space is meant the norm of their difference: x − y, x − y . x−y = In other words, the norm of a point in a pre-Hilbert space is equal to its distance from zero. 1 (Schwarz inequality). 1) x, y ∈ X.