---- LINEAR SYSTEMS: -- write a system of linear equations in matrix form -- decide if a given system has a solution; -- solve a linear system by Gauss- Jordan elimination or by finding rref of the matrix -- interpret the linear system geometrically in simple cases (2 or 3 variables only; equations give lines on the plane or planes in the 3-space); make conclusions on how many solutions the system has --- MATRICES AND VECTORS -- Compute dot product of vectors (if defined); use dot product to determine if two vectors are perpendicular; find orthogonal projection of a given vector to a given line. -- Compute product of a matrix and a vector (if defined) express this product as a linear combination of the columns of the matrix -- compute product of two matrices (if defined); interpret matrix multiplication as composition of corresponding linear transformations -- Determine if a given n x n matrix is invertible; find inverse matrix --- SUBSPACES, LINEAR INDEPENDENCES, BASIS, DIMENSION -- decide if given vectors are linearly independent (SUBSPACES WILL TYPICALLY BE GIVEN AS A SPAN OF VECTORS or AS KER or IM of a linear transformation.) -- Decide if a given vector belongs to a given subspace; -- Decide if given collection of vectors forms a basis of a given subspace; -- Find a basis of a given subspace -- Find dimension of a given subspace -- Basis change: find coordinates of a given vector with respect to a (new) basis of a (sub)space. ---- LINEAR TRANSFORMATIONS --- Write a matrix of a linear trasformation given geometrically (projections, reflections, reflections and scaling in R^2 and R^3; simple cases ONLY) --- Interpret geometrically a linear transformation given by a matrix (simple cases in R^2 and R^3 ONLY) --- Find Kernel and Image for a given linear transformation --- Use relation between dimensions of kernel, image, and the domain space (RANK-NULLITY THEOREM) to find dim Ker from Dim Im abd vice versa --- Find bases for Ker and Im --- use information on kernel and image to make conclusions about the number of solutions for corresponding linear systems --- basis change: write a matrix of a given transformation with respect to a new basis; use a convenient basis & a basis change to write a matrix of a transformation given geometrically (projections and reflections onto arbitrary line on the plane) --- RANK OF A MATRIX/LINEAR TRANSFORMATION -- interpret and compute rank from rref of a matrix -- interpret and compute rank as dimension of image of linear transformation -- understand possible values of rank for square and rectangular matrices -- using rank to make conclusions about invertibility of a square matrix, and about the number of solutions of a linear system (with a square or non-square coefficient matrix) ----LINEAR SPACES (AKA VECTOR SPACES)---- --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. --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 --Find a basis of a given subspace, determine its dimension --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) --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). -- write the matrix of a given transformation T: L --> L in a given basis. ---- DOT PRODUCT, ORTHOGONALITY IN R^n ---- -- 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 -- 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!), decomposing a vector as the sum of its projection onto the subspace and a vecotr orthogonal to the subspace; finding length of vector from its coordinates in an orthonormal basis; -- 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) -- 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 --- -- 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) -- use determinants to find out whether a matrix is invertible; formula for the inverse matrix of a 2x2 matrix (with non-zero det) -- 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) -- Geometric meaning of the determinant: find areas/volumes of paralellograms/parallelepipeds formed by the given vectors ----EIGENVALUES AND EIGENVECTORS -- check if a given number is an eigenvalue, a given vector is an eigenvector -- Compute eigenvalues for a given matrix -- for each eigenvalue, find an eigenvector; find the eigenspace corresponding to the eigenvalue, determine the dimension of this eigenspace -- determine if the matrix is diagonalizable; if so, find the diagonal form of the matrix and specify the corresponding basis.