By Edwards, Charles Henry
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Those notes shape the contents of a Nachdiplomvorlesung given on the Forschungs institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. ok. Chandrasekharan and Prof. Jurgen Moser have inspired me to write down them up for inclusion within the sequence, released via Birkhiiuser, of notes of those classes on the ETH.
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6 Two vector spaces V and W are called isomorphic if and only if there exist linear mappings S : V → W and T : W → V such that S T and T S are the identity mappings of W and V respectively. Prove that two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension. 7 Let V be a finite-dimensional vector space with an inner product , . The dual space V* of V is the vector space of all linear functions V → . Prove that V and V* are isomorphic. Hint: Let v1, . . , vn be an orthonormal basis for V, and define by θj( vi) = 0 unless i = j, θj( vj) = 1.
An are as usual the column vectors of A. Then (4) above says that the determinant of A is multiplied by r if some column of A is multiplied by r, (5) that the determinant of A is unchanged if a multiple of one column is added to another column, while (6) says that the sign of det A is changed by an interchange of any two columns of A. ” (II) The determinant of the matrix A is equal to that of its transpose At. The transpose At of the matrix A = (aij) is obtained from A by interchanging the elements aij and aji, for each i and j.
5 is equivalent to the following theorem: Suppose that the equations have only the trivial solution x1 = · · · = xn = 0. Then, for each b = (b1, . . , bn), the equations have a unique solution. Hint: Consider the vectors aj = (a1j, a2j, . . , anj), j = 1, . . , n. 7 Verify that any two collinear vectors, and any three coplanar vectors, are linearly dependent. 3 INNER PRODUCTS AND ORTHOGONALITY In order to obtain the full geometric structure of n (including the concepts of distance, angles, and orthogonality), we must supply n with an inner product.
Advanced calculus of several variables by Edwards, Charles Henry