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HARVARD MATH 21B - Review for the First Mid-Term

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Review for the First Mid-Term of Math 21bOctober 24, 2007• Coefficient matrix and augmented matrix of a system of linear equa-tions.• Reduced row-echelon form of a matrix characterized by three condi-tions:a. If a row has nonzero entries, then the first nonzero entry is 1, calledthe leading 1 in this row.b. If a column contains a leading 1, then all other entries in that columnare zero.c. If a row contains a leading 1, then each row above contains a leading1 further to the left.• Reduction of a matrix to reduced row-echelon form by using three kindsof row operations (swapping rows, multiplying a row by a nonzero num-ber, and adding a multiple of a row to another).• Use reduction to reduced row-echelon form to determine whether asystem of linear equations is inconsistent, uniquely solvable, or solvablewith an infinite number of solutions and to give a general solution (ifit exists) by using free variables.• Use reduction to reduced row-echelon form to determine whether asquare matrix is invertible and to find its inverse if it is invertible.• Determinant of a 2 × 2 matrix. Formula for the inverse matrix of a2 × 2 matrix with nonzero determinant.• The span of a set of vectors. Redundant vectors in a sequence of vectors.Linear dependence and independence of a set of vectors. Subspaces.Bases. Dimension.• Determine the rank and the nullity of a matrix. Find a basis for theimage and for the kernel of a matrix. Rank-Nullity Theorem.1• Equivalence conditions for the invertibility of an n × n matrix A:unique solvability of A~x =~b, rref(A) = In, rank(A) = n, im(A) = Rn,ker(A) = {~0}, column vectors forming a basis of Rn, column vectorsspanning Rn, column vectors linearly independent.• Special linear transformations: rotations, dilations, projections (onto aline or a plane), reflections, and shears (horizontal and vertical).• The column vectors of the matrix of a linear transformation equal toits images of the standard vectors.• Relation between matrix multiplication and the composition of lineartransformations.• Coordinates with respect a basis of a subspace. Matrix of a lineartransformation with respect to a basis. Relation of matrices of thesame linear transformation with respect to two different bases. Similarmatrices. Powers of similar matrices. Similarity as an equivalencerelation.• Concept a linear space (also known as a vector space). Addition andscalar multiplication in a linear space and the laws (associativity, com-mutativity, distributivity, etc.) satisfied by them. Examples of linearspaces: solutions of differential equations, spaces of polynomials, spacesof matrices, etc. Dimension of linear space. Finite and infinite dimen-sion.The First Midterm covers up to and including Section 4.1of Bretscher’s book on Linear Algebra with


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HARVARD MATH 21B - Review for the First Mid-Term

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