Math 33A with ProfessorSection with Sam XuApril 12, 2012• Email: (myfullnamenospaces)@math. .eduOffice hours: inThe Student Math Center (SMC): Mondays through Thursdays 9am to 3pm in MS 3974.I am there on from to .• About the discussion section:– We will start with homework discussion.– Please have at least one question and/or discussion topic ready.– If there is time left, we will discuss some material in more depth.• Classroom Etiquette:– Please turn off cell phones.– Please be quiet during the section.– Please stay for the entire length of the discussion section.• Recommendations specific to the course– Overarching principle: Prioritize mastery of row reduction (and get fast at it) and the conceptualconnections.– This is the first serious conceptual math class. So, experiment with each problem and see howother ideas fit in:∗ For each computational exercise, try to generalize. E.g., Can you replace the specific vec-tors [1 3 9] with general vectors in Rn? How general is each computation: i.e., for how whichtypes of matrices (diagonal, elementary, lower triangular, orthogonal, positive definite, invert-ible, shear transformations, permutation, square/non-square, etc) will the given technique beapplicable?∗ For each conceptual question, try to connect to previous ideas: this may be your first coursein which there are many new abstract ideas that build upon themselves.– After finishing the homework, work through the course notes (and possibly some section notes)carefully. Possible things to keep in mind: 1) connect ideas with previous notes and homeworks2) possibly being able to lecture on it without the aid of notes.– Some supplementary material include Schaum’s Outline on Linear Algebra ($15 to $20 in thebookstore).1NotesDot ProductsTwo of the central properties of dot products are its definition and an immediate corollary:Let ~v = (v1, . . . , vn) and ~w = (w1, . . . , wn) be vectors in Rn. We define their dot products as~v · ~w = v1w1+ . . . + vnwn. (1)From here one can show that~v · ~w = k~vkk ~wk cos θ (2)where θ is the angle between the two vectors.Matrix ProductsKeep in mind that if A is an n × m matrix and B is a k × ` matrix, then we must have m = k for the matrixproduct AB to make sense. In this case AB is a n × ` matrix.An Introduction to LU FactorizationHere we shall introduce the concept of LU factorization via several examples. To begin, let us consider thesystem of linear equations3x + 7y = 56x + 2y = 3,whose matrix form is3 76 2xy=53.Using usual methods, we would solve this equation by subtracting twice the first row from the second row.This can be encoded as3x + 7y = 56x + 2y = 3R2→R2−2R1−→3x + 7y = 50x − 12y = −7One can then solve the system: first for y and then for x.In matrix form, this can be realized by multiplying by a lower triangular matrix on the left:3 76 2xy=53−→1 0−2 13 76 2xy=1 0−2 153−→3 70 −12xy=5−7After doing this computation, this gives us two pieces of information: a solution to the linear system asdescribed above as well as the LU factorization of the matrix A =3 76 2. Namely, with U =3 70 −12and L−1=1 0−2 1, we can show that L =1 02 1and thus one can checkA = LU.Here is an introduction to elementary matrices, as a buzzword. Let us make some observations here.2• The elementary matrix1 0−2 1represents the row operator “subtract twice the first row from thesecond row.” To wit, observe that1 0−2 1a bc d=a bc − 2a d − 2b.Thus, it should make sense that the inverse operation should be “add twice the first row to the secondrow”. In matrix language:1 0−2 1−1=1 02 1.Using pencil and pencil, check that the matrices do indeed multiply to the 2 × 2 identity matrix.• The elementary matrix1 00 −112represents the row operator “multiply the second row by −112”. Towit, observe that1 00 −112a bc d=a b−112c −112d.Thus, it should make sense that the inverse operation should be “multiply the second row by −12”.In matrix language:1 00 −112−1=1 00 −12Using pencil and pencil, check that the matrices do indeed multiply to the 2 × 2 identity matrix.• Occasionally, one may need to swap two rows. The elementary matrix is given by0 11 0. To wit,observe that0 11 0a bc d=c da b.Note that doing a swap twice gets us back to the original position. Namely,0 11 0−1=0 11 0Using pencil and pencil, check that the matrices do indeed multiply to the 2 × 2 identity matrix.Let us consider a slightly more complicated example:x + 2y + z = 32x + 6y + 5z = 45x + 10y + 13z = 5,So that its corresponding matrix equation is1 2 12 6 55 10 13xyz=345.This can be solved via the row reductionsx + 2y + z = 32x + 6y + 5z = 45x + 10y + 13z = 5R2→R2−2R1−→x + 2y + z = 30x + 2y + 3z = −25x + 10y + 13z = 5R3→R3−5R1−→x + 2y + z = 30x + 2y + 3z = −20x + 0y + 8z = −103The matrix realization can be given as follows. The coefficient matrix is A =1 2 12 6 55 10 13. Our first rowreduction operation was to subtract twice the first row from the second row, so (see the bullet points above)we use the matrix1 0 0−2 1 00 0 1. Our second row reduction operation was to subtract five times the first rowfrom the third row, so we use the matrix1 0 00 1 0−5 0 1. The realization in terms of matrices is then1 2 12 6 55 10 13xyz=345⇐⇒1 0 00 1 0−5 0 11 0 0−2 1 00 0 11 2 12 6 55 10 13xyz=1 0 00 1 0−5 0 11 0 0−2 1 00 0 1345.For the left hand side, we multiply all the matrices together and for the right hand side, we multiplyeverything to get1 2 10 2 30 0 8xyz=3−2−10.As above, this computation gives us the LU factorization: U =1 2 10 2 30 0 8and L−1=1 0 00 1 0−5 0 11 0 0−2 1 00 0 1,which means L =1 0 02 1 00 0 11 0 00 1 05 0 1=1 0 02 1 05 0 1. Using pencil and paper double check that we indeedhave LU = A.Projections on to a HyperplaneConsider a non-zero vector ~v = (v1, . . . , vn0 in Rn. Define the n − 1 dimensional hyperplane in Rnby