Date | Material covered |
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1/3 | Announcements. What the class is and what it isn't. Introduction: linear algebra is about vector spaces and linear maps. The first week or two will be about vector spaces, the rest of the class about linear maps. Properties of R and C: they are both fields (i.e. satisfy the commutativity, associatity, identity, additive inverse, multiplicative inverse, and distributive properies listed for C on page 2-3.) In Axler's book, F denotes the real or complex numbers, but we shall work a little more generally and let it be any field. Then I gave the first main definition of the class: the axioms of a vector space over F. Stated and proved some elementary properties (page 11-12). Gave examples. |
1/5 | The first homework set is posted. More examples of vector spaces. Linear maps (definition in Ch. 3). Subspaces of a vector space. Sum of subspaces. Direct sum of subspaces: If U_1, ..., U_m are subspaces of a vector space V we say that "their sum is direct" if the only way to write 0 = u_1 + ... + u_m with u_i in U_i for all i, is to have all u_i be 0. |
1/7 | Criteria for sums being direct: I stated (and proved) modified versions of propositions 1.8 and 1.9 in the book, which I called proposition 1.8' and 1.9'. In 1.8' I omitted assumption (a), and can only conclude that the sum is direct. In proposition 1.9' I omitted the assumption that U+W=V, and can only conclude that the sum is direct. Linear combinations. Span of a list of vectors. Finite dimensional vector spaces. Example F^n is finite dimensional. |
1/10 | Linear independence. Linear dependence. Basis. Proved the "linear dependence lemma" and the (very important) theorem 2.6: Any spanning list of vectors in V is at least as long as any linearly independent list. |
1/12 | Two bases for the same vector space V has the same number of elements. Any finite dimensional space has a basis. In fact, we can obtain a basis by discarding vectors from any given spanning list. Definition of dimension: the number of elements in a basis. Any linearly independent list in a finite dimensional vector space can be extended to a basis. The dimension of a subspace of a finite dimensional vector space V is at most as large as the dimension of V. Criteria for lists to be bases (proposition 2.16 and 2.17): If V has dimension n, then any linearly independent list of length n is a basis, and any spanning list of length n is a basis. |
1/14 | The dimension of a sum. Criteria for sums being direct. Linear maps (definition and a few examples). |
1/19 | More examples of linear maps. The set of linear maps from V to W is itself a vector space. Composition of linear maps. Null space and Range of a linear map. Dimension formula. Injective and surjective maps, and the relation to injectivity and surjectivity. |
1/21 | Invertible linear maps. Isomorphic vector spaces. Dimension criteria: Finite dimensional vector spaces V and W are isomorphic if and only if dim(V) = dim(W). In that case, a linear map T:V -> W is invertible if and only if it is injective or surjective (i.e. if the dimensions agree, it suffices to check either surjectivity or injectivity; the other is then automatic). |
1/24 | Matrices: Definition of Mat(m,n,F). Vector space structure on Mat(m,n,F). The matrix M(T) of a linear map T: V -> W with respect to bases of V and W. L(V,W) is isomorphic to Mat(m,n,F) when V and W are finite dimensional, V has dimension n and W has dimension m. In particular, L(V,W) is finite dimensional, and dim(L(V,W)) = (dim(V))(dim(W)). Multiplication of matrices. The matrix of a composition: M(TS) = M(T)M(S), i.e. "the matrix of TS is the product of the matrix of T and the matrix of S". |
1/26 | The matrix of a vector. M(Tv) = (M(T))(M(v)). Review of polynomials (following Axler Ch 4, covering only the part about complex numbers). Invariant subspaces. Eigenvalues and eigenvectors. |
1/28 | Eigenvalue and eigenvectors. lambda is an eigenvalue of T if and only if Null(T-lambda I) is non-zero. Eigenvectors corresponding to different eigenvalues are linearly independent. An operator on V has at most dim(V) distinct eigenvalues. Polynomials evalutated on operators. |
1/31 | Every linear operator on a finite dimensional non-zero complex vector space has an eigenvalue. Upper triangular matrices. For every operator T on a finite dimensional complex vector space V, there is a basis of V with respect to which the matrix of T is upper triangular. |
2/2 | Class taught by Pete Storm. |
2/4 | Finished proof of proposition 5.21. Example of a linear transformation T in L(V) such that V does not have a basis consisting of eigenvectors of T, namely V = C^2, T(x,y) = (y,0). Started Chapter 6 about inner product spaces: Some motivation about the dot product in R^n. Basic definitions. |
2/7 | Inner product spaces: immediate consequences of the axioms. Orthogonal decomposition. Cauchy-Schwarz' inequality. The triangle inequality. |
2/9 | Orthonormal lists and their properties. Orthonormal bases. Gram-Schmidt. As a corollary, every finite dimensional inner-product space has an orthonormal basis. |
2/11 | Orthogonal complement. Orthogonal projection. Formula for orthogonal projection onto U, given orthonormal basis of U. Every linear operator T on a finite dimensional inner product space V has upper triangular matrix with respect to some basis of V. |
2/14 | Interpretation of Gram-Schmidt formula: In the induction step we subtract from v_{j} its orthogonal projection onto the span of the previous vectors. Orthogonal projection onto U gives the nearest point. Functionals. Adjoints of linear maps. |
2/16 | Properties of adjoints. Matrices of linear maps between inner product spaces. With respect to orthonormal bases, the matrix of the adjoint of T is the conjugate transpose of the matrix of T. Self-adjoint operators. Normal operators. Introduction to the spectral theorem. |
2/18 | Polarization identity. If T is self-adjoint, then T=0 if and only if <Tv,v> = 0 for all v. If T is normal, then eigenvectors corresponding to distinct eigenvalues are orthogonal. Statement of the (complex) spectral theorem. |
2/21 | President's Day, no lecture. |
2/23 | Proof of the complex spectral theorem. Statement of the real spectral theorem and warmup to its proof: real polynomials. |
2/25 | Any self-adjoint operator has an eigenvalue. Proof of the spectral theorem for operators in real inner product spaces. |
2/28 | Generalized eigenvectors. The nullspace of (T- lambda I) to the jth power is the nullspace of (T- lambda I) to the (dim V)th power, if j is greater than or equal to dim V. Nilpotent operators. Stated theorem 8.10 and began the proof. |
3/2 | Proof of theorem 8.10. Multiplicity of an eigenvalue, defined as the dimension of the null space of (T - lambda I) to the power dim V. Theorem 8.10 shows that the multiplicity equals the number of times lambda occurs on the diagonal of M(T), if M(T) is upper triangular. Example: V is the polynomials of degree at most 3, T is differentiation. We calculated the multiplicity of 0 (it is 4) in two ways: from the definition, and from an upper triangular matrix. |
3/4 | Sum of multiplicities equals the dimension of the vector space. The Cayley-Hamilton theorem. |
3/7 | Decomposition theorem (thm 8.23): If T is an operator on V, then V is the direct sum of subspaces U_i, where U_i is the null space of (T - lambda_i I) to the power dim(V). Block diagonal matrices and the matrix of a decomposed operator. Theorem 8.28: for any operator on a complex vecttor space V, there exists a basis consisting of generalized eigenvectors, such that the matrix of T is block diagonal, and each block is upper triangular and has the same number on the diagonal. Definition of minimal polynomial. |
3/9 | Minimal polynomial: The minimal polynomial of T divides a polynomial p, if and only if p(T)=0. By Cayley-Hamilton, the minimal polynomial divides the characteristic polynomial. Jordan normal form: proof in the case T is nilpotent. |
3/11 | Last lecture! Finish proof of Jordan normal form. Say goodbye and wish good luck with exam. |