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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.

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