LINEAR ALGEBRA STUDY GUIDE -- 1998/05/04, rusin@math.niu.edu Here is a checklist you might want to use to prepare for the Math 240 exam. References refer to Kolman's text, but the test is on the material, not the book! I. Algorithms. We have learned only two! Make sure you can do them. ROW REDUCTION to reduce a matrix to reduced row-echelon form GRAM-SCHMIDT PROCESS to replace a set of vectors with an orthonormal set spanning the same subspaces Do not expect to see any exam questions which specifically ask you to perform these algorithms! However, you will need one or both to complete all the other computations we discussed this semester: II. Computational problems. In every case you should be able to explain why your methods compute what you claim they compute, how you read off the required information from a matrix, etc. A. Given a system of linear equations, determine whether it has any solutions, and if so, a description of the solution set. B. Given a matrix A, compute transpose(A), ker(A), im(A), rowspace(A), rank(A), nullity(A), reduced-row-echelon-form(A), also column, also row-and-column (p. 65) if A is square: tr(A), det(A), A^(-1), adj(A), charpoly(A), eigenvalues(A), eigenvectors(A), factorizations: A=: product of elementary, P J Q (p.67) P-inverse D P (Chap. 6); you can skip the LU and QR factorizations C. Given a set of vectors S in a vector space V, compute a basis for W=span(S), an orthonormal basis for W, dim(W) D. Given a vector v in a vector space V and a basis B of V, compute coordinate vector [v]_B (a column vector!) E. Given two bases of V, compute two change-of-basis matrices P (one each way) F. Given a vector v in an inner product space V, compute length(v), angle(v,w) and dist(v,w) (w = another vector), proj_W(v) (W a subspace of V) G. Given an inner product space and a basis for it, compute the matrix representing the inner product w.r.t. this basis. H. Given a linear transformation L : V -> W, compute ker(L), im(L), rank(L); given bases for V and W, compute the matrix representation of L You should also be able to compute various quantities straight from a definition, e.g. you should be able to compute a matrix product, a dot product of vectors, etc. You should be able to perform more than one of these calculations at a time, e.g. to compute an orthonormal basis of the kernel of a linear transformation. You should be able to perform these calculations even if some entries of a matrix are variables. If the numbers given you are integers, you should be able to express your results as fractions or radicals, not (only) decimals. You should not have to assume that the vector spaces are R^n, that the bases are some particular "natural" bases, etc.; of course, those examples are very common, so you should practice them in particular -- just not exclusively. If you can perform all these calculations flawlessly you should be able to get at least 50% of the possible points on the final. However, you cannot expect to get any more than that if you learn only the computations! Here are some other things expected of you: III. Definitions. You should be able to give a fairly concise definition of each of the following. Your definition should be in complete sentences; heavy use of symbolism is fine IF the symbols, when read aloud exactly as they appear, form complete sentences too. You should know the definition accurately enough so that if we give you an object and say, "Is this a XXX?", you can answer either "No, it fails part YYY of the definition" or "Yes, here is a demonstration that all the parts of the definition are satisfied." You should know the definition instinctively enough that if a problem presents a situation by using one of these words, you should be able to restate the problem in more elementary language -- without the word. Nouns: You should be able to define all the quantities you were asked to compute in II; in particular, you should be able to define "rank", "basis", "kernel", "coordinates",... Other key words and phrases: vector space, subspace, isomorphism, linear transformation, inner product. Other nouns have occurred frequently, and I think most people know these: "matrix", "product (of matrices)", "row operation", "linear combination", "standard basis", etc. A number of terms occured briefly or in examples ("cofactor", "continuous", "multiplicity",...) Adjectives: Said of a matrix: upper-triangular, symmetric, diagonal, elementary, (non-)invertible, (non-)singular, positive definite, diagonalizable, orthogonal Said of pairs of matrices: equal, (row-, column-, )equivalent, similar Said of a function: one-to-one, onto, invertible, linear Said of a set of vectors: linearly (in)dependent, spanning, closed under (addition, scalar multiplication) For 2 vectors: parallel, perpendicular, orthogonal, Said of a vector space: finite-dimensional Said of pairs of vector spaces: isomorphic, orthogonal Verbs: A set of vectors might: span (a vector space) A matrix might: represent (a linear transformation, or an inner product; in each case bases must be given) You might be asked to: row-reduce, diagonalize (a matrix). Other features of language: please learn to spell and pronounce the singular and plural forms of matrix, basis, kernel, determinant, indeterminate, theorem, scalar, (in)dependent, invertible, commutative, separate, homogeneous, symmetric A mastery of the terms and language of linear algebra includes knowing the notation. Much of what you need is on the inside cover of the text. Note that e.g. "0" might refer to a number, a vector, or a matrix. Also note that some pieces of notation are to be thought of as functions, e.g. det(A) or [v]_S. Finally note that notation is not always standard; be prepared to see the entries of a matrix A written either a_ij or A_ij ; inner products might be (v,w) or ; transposes might be A^T or A^t (or with the t or T superscript on the left of A). Please be able to WORK WITH the notation; for example, you should be able to prove SYMBOLICALLY that the product of two diagonal matrices is diagonal ("D=diagonal" means D_ij = 0 if i doesn't equal j so...) If you can perform all the calculations and also apply all the definitions, you should be able to answer about three-quarters of the test questions without hesitation. The remaining parts of the questions will ask you to apply what you have learned about these concepts and make deductions. You will be expected to explain how you know your deductions are valid (that is, you will give proofs). Any proof which is more complicated than a computation or a check of a definition will require referring to one or more of the: IV. Major results There are a number of major theorems we have learned this semester, and many minor but interesting ones. In general, we will not ask you simply to state the theorem, but you will need to apply it, which requires knowing with precision what the assumptions of the result are. I am hoping that by this point in the semester many of these results have become so ingrained in your mind that you hardly think to mention them anymore when using them. For example, it's OK to use without quoting them all the theorems which prove that the calculations of section II really do what they say they do, if you show your intermediate steps. Here are some results you may want to review: learn what they say, look at the illustrative examples, see how the theorem gets applied, look at the homework problems which required them. Theorem 1.1-1.2-1.3-1.4: properties of matrix algebra. Theorem 1.5-...-1.8: properties of matrix inverses. Theorem 1.18: singularity detected by row operations. Theorem 2.2, 2.6 etc.: elementary properties of vector spaces p. 144: role of change-of-basis matrix for coordinate vectors Corollary 2.1: Invariance of Dimension (and its corollaries) Theorem 2.13, 2.15: Isomorphism with R^n Theorem 2.17: Equality of several notions of rank Theorem 2.18: rank+nullity=#columns P. 169: rank and invertibility Theorem 3.3: Cauchy-Schwartz inequality Corollary 3.1: Triangle inequality Theorem 3.4: orthogonality and independence (and coordinates) Theorem 4.1 et seq: elementary properties of linear transformations Theorem 4.5: rank+nullity=dim(V) Theorem 4.8: computation of L(v) via representation matrix Theorem 4.11: role of change-of-basis matrix for linear transforms Theorem 4.13-4.14: interpretation of similarity Theorem 5.1-5.7: elementary properties of determinants Theorem 5.8 det=0 means singular Theorem 5.9, Cor. 5.4: multiplicativity of determinants Theorem 5.10: expansion of det(A) by cofactors Theorem 5.12: determinants, adj(A), and inverses Theorem 6.1-6.4: diagonalizability and eigen-stuff Theorem 6.5: invariance under similarity Theorem 6.6-6.9: diagonalization of symmetric matrices