----LINEAR SPACES (AKA VECTOR SPACES)---- (L1) determine whether a given collection of some elements (with operations of addition and scalar multiplication) forms a vector space. Typical examples are matrices or functions with particular properties. Determine whether a given "part" (typically described by some property) of a vector space is a subspace. (L2) determine whether given "vectors" (in a general linear space) is linearly independent; whether they belong to a particular subspace, whether they span the subspace, whether they form a basis of the subspace (L3) Find a basis of a given subspace, determine its dimension (L4) Use coordinate transformation (with respect to a given basis) to make an isomorphism between the given linear space and R^n for the appropriate n. Find coordinates of a given "vector" in a given basis. Use the coordinate transformation together with argument in R^n to answer questions L2,L3 above. (L2,L3 can also be answered directly, working in the original vector space) (L5) Determine whether a given transformation is linear. Find its image and kernel, their bases and dimensions. (Similarly to the R^n case, you can use the Rank-Nullity theorem to find dim Ker from Dim Im and vice versa). Determine whether a given linear transformation is an isomorphism (use flow chart on p.183 or similar arguments). (L6) write the matrix of a given transformation T: L --> L in a given basis. (L7) Work with the inner product in the space of functions (defined by = integral of f(t)g(t), see 5.5): determine whether two functions are orthogonal, find orthonormal bases, find orthogonal projection onto a give subspace, etc. (Compare O1, O2 below.) ---- DOT PRODUCT, ORTHOGONALITY IN R^n ---- (O1) compute dot product of two vectors (if defined), determine whether two vectors are orthogonal, find angle between two vectors, find length of a given vector (O2) Properties of orthonormal vectors (orthonormal vectors are linearly indep); finding coordinates of a given vector with respect to *orthonormal* basis (using dot product is faster & easier than material from Ch. 3!); finding an orthogonal projection of a vector onto a given subspace (when an orthonornal or an orthogonal basis is given!); finding length of vector from its coordinates in an orthonormal basis (O3) Gram-Schmidt process: apply to a given collection of lin. indep. vectors to obtain an orthonormal basis of the (sub)space; illustrate the process by a picture (in simple cases) (O4) the concept and properties of orthogonal transformation: orthogonal transf. preserves angles and length. Determine whether the given transformation/matrix is orthogonal using 1) its geometric properties, 2) checking that the columns are orthonormal, 3) A A^T =I. ---- DETERMINANTS --- (D1) Compute determinants by different methods: 1) special rules for 2x2 and 3x3 matrices; 2) sum of products (one entry from each row/column) with signs (don't forget the signs!) - works well for matrices with a lot of zeroes; 3) reducing to a simpler matrix via elimination by using properties (stated in D3) (D2) use determinants to find out whether a matrix is invertible; formula for the inverse matrix of a 2x2 matrix (with non-zero det) (D3) Properties of determinants: linearity in row/column (thm 6.2.2), elementary row operations (Thm 6.2.3), transpose (thm 6.2.1), determinant of a matrix with 2 equal rows/columns (or a row/column of zeroes) is zero, determinant of a product (thm 6.2.6) (D4) Geometric meaning of the determinant: find areas/volumes of paralellograms/parallelepipeds formed by the given vectors ========================================================================================= REMARKS: there will be three multi-part questions on the test, following the three large topics above. (Only a selection of tasks listed above will appear, as time permits.) Note that 1) some of the questions may include a parameter (for what value(s) of parameter(s) are these vectors orthogonal? For what values of parameter(s) is this matrix invertible? For what values of parameter(s) is this matrix orthogonal? For what value of parameters are these functions linearly independent? etc) 2) A subspace (in questions such as L2, L3, O2, O3) can be given as a span of several vectors, as a kernel/image of a linear transformation, by some condition such as p(1)=0, by an equation such as 2x -7 y+z=0. 3) Properties of determinants, orthogonal matrices, etc can be tested via "theoretical" questions such as 11-16 on p.289, 38 on p.38, 43-44 p.276, 5-11, 31 on p.233, 29 p.217.