Solutions Manual for Lang's Linear Algebra

I Vector Spaces.-
1. Definitions.-
2. Bases.-
4. Sums and Direct Sums.- II Matrices.-
1. The Space of Matrices.-
2. Linear Equations.-
3. Multiplication of Matrices.- III Linear Mappings.-
1. Mappings.-
2. Linear Mappings.-
3. The Kernel and Image of a Linear Map.-
4. Composition and Inverse of Linear Mappings.-
5. Geometric Applications.- IV Linear Maps and Matrices.-
1. The Linear Map Associated with a Matrix.-
2. The Matrix Associated with a Linear Map.-
3. Bases, Matrices and Linear Map.- V Scalar Products and Orthogonality.-
1. Scalar Products.-
2. Orthogonal bases, Positive Definite Case.-
3. Application to Linear Equations; the Rank.-
4. Bilinear Map and Matrices.- 5. General Orthogonal Bases.-
6. The Dual Space and Scalar Products.-
7. Quadratic Forms.-
8. Sylvester's Theorem.- VI Determinants.-
2. Existence of Determinants.-
3. Additional Properties of Determinants.-
4. Cramer's rule.-
5. Triangulation of a Matrix by Column Operations.-
6. Permutations.-
7. Expansion Formula and Uniqueness of Determinants.-
8. Inverse of a Matrix.-
9. The Rank of Matrix and Subdeterminants.- VII Symmetric, Hermitian and Unitary Operators.-
1. Symmetric Operators.-
2. Hermitian Operators.-
3. Unitary Operators.- VIII Eigenvectors and Eigenvalues.-
1. Eigenvectors and Eigenvalues.-
2. The Characteristic Polynomial.-
3. Eigenvalues and Eigenvectors of Symmetric Matrices.-
4. Diagonalization of a Symmetric Linear Map.-
5. The Hermitian Case.- IX Polynomials and Matrices.-
2. Polynomials of Matrices and Linear Maps.- X Triangulation of Matrices and Linear Maps.-
1. Existence of Triangulation.-
3. Diagonalization of Unitary Maps.- XI Polynomials and Primary Decomposition.-
1. The Euclidean Algorithm.-
2. Greatest Common Divisor.-
3. Unique Factorization.-
4. Application to the Decomposition of a Vector Space.-
5. Schur's Lemma.-
6. The Jordan Normal Form.- XII Convex Sets.-
4. The Krein-Milman Theorem.- APPENDIX Complex Numbers.