USA MA 237 - Linear Algebra Study Guide For The Final

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Dr. Byrne Math 237Fall 2009 Section 1Linear Algebra Study GuideFor The FinalDate and Time of Final Exam: Monday, December 7th, 8:00 am – 10:00 amThe final exam is cumulative and will focus on the computational skills you have acquired in this course. This study guide consists of reviewing 8 linear algebra skills. These skills are worth mastering not only because they are important in linear algebra but also because at least 80% of the final will be based upon them. (The remaining 20% will come from previous exam questions and emphasized class notes.) My office hours will be 10am-11am as usual, on all official university working days until the final. The skills are:1. Identifying the Span of A Set of Vectors in 2 and finding the basis for the span of a set of vectors.2. Row-reducing a matrix without a calculator.3. Describing the solution to a linear system by writing the solution in parametric form.4. Finding the determinant of a matrix without using a calculator.5. Applying a linear transformation to a given vector. 6. Defining a linear transformation by its effect on given set of vectors.7. Finding the eigenvalues, eigenvectors and eigenspace of a square matrix without acalculator.8. Identifying whether or not a given square matrix is diagonalizable.Key DefinitionsLinear combination: A linear combination of n vectors v1 , v2, …, vn is a vector w where w=a1v1 + a2v2 + …+ anvn for scalars a1, a2, …, an.Linear dependence of a vector on a set: A vector w is linearly dependent on v1 , v2, …, vn if w is a linear combination of v1 , v2, …, vn.Linear dependence of a set: A set S is linearly dependent if it contains at least one vectorthat is linearly dependent on the others.Span of a set: Given a set S, the set of all linear combinations of the elements of S is the span of S. Subspace: A non-empty set W is a subspace of a vector space V if W is a subset of V and W is closed under linear combinations (i.e., any linear combination of the elements of W are also contained in W).Row Space: The row space of a matrix A is the span of the rows of A.Column space: The column space of a matrix A is the span of the columns of A.Note: row reduction changes the column space but does not change the row space. Basis: A basis of a vector space V is a set of vectors S belonging to V such that (1) S spans V and (2) S is linearly independent.Dimension: The dimension of a vector space V is the number of vectors contained in a basis of V. Rank: The rank of a matrix A is the dimension of the row space, which is also equal to the dimension of the column space of A.Nullspace: The nullspace of a matrix A is the span of the solutions to Ax=0.Nullity: The nullity of a matrix A is the dimension of the nullspace.Consistent linear system: A system of linear equations is consistent if it has at least one solution.Homogeneous linear system: A system of linear equations Ax=B is homogeneous if B=0.Singular matrix: A square matrix A is singular if the equation Ax = 0 has a nonzero solution for x. Nonsingular matrix: A square matrix A is nonsingular if the only solution to the equation Ax = 0 is x = 0.Eigenvalue: A scalar  is an eigenvalue of A if there exists a vector x such that Ax=x. Characteristic polynomial: The characteristic polynomial of an n by n matrix A is the polynomial in  given by the formula det(A - I).Algebraic multiplicity of an eigenvalue: The algebraic multiplicity of an eigenvalue c of a matrix A is the number of times the factor (-c) occurs in the characteristic polynomial of A.Geometric multiplicity of an eigenvalue: The geometric multiplicity of an eigenvalue c of a matrix A is the dimension of the eigenspace of . Defective (deficient) matrix: A matrix A is defective if A has an eigenvalue whose geometric multiplicity is less than its algebraic multiplicity.Linear transformation: A linear transformation from V to W is a function T from V to Wsuch that: 1. T(u+v) = T(u) + T(v) for all vectors u and v in V; and 2. T(av) = aT(v) for all vectors v in V and all scalars a.1A. Identifying the Span of A Set of Vectors in  2 a) The span of a single vector is a line (or a point in the case of [0,0]). Describe the span of the vector v=12.i) List the coordinates of 5 distinct points thatare in the span of12.A=D=B=E=C=ii) Clearly plot and label each of the five points A, B, C, D, E if they are in the range of the provided graph.iii) Write down the equation of the line that is the span of12.b) The span of two vectors depends upon whether they are linearly independent or not. Describe the span of the set of vectors21,1211vvS.i) List the coordinates of 6 distinct points thatare in the span of S.A=D=B=E=ii) Clearly plot and label each of the six points A, B, C, D, E, F if they are in the range of the provided graph.C=F=iii) What is the span of S geometrically?1B. Finding the basis for the span of a set of vectors.Consider a basis B of the span S of a set of vectors {v1 , v2, …, vn}. By the definition of a basis, B must span S and also B must be linearly independent. The set of vectors {v1 , v2, …, vn} already span their span (thus they span S), so we just need to find a subset of {v1 ,v2, …, vn} that is linearly independent but which has the same span (i.e., a linearly independent subset with the same dimension.) We do this by finding the rank of a matrix composed of the vectors v1 , v2, …, vn. 1. Find a basis for the span of 211211,000241,221000,221241.2. Find a basis for the span of 777,737,010,202,101,333.2. Row-reducing a matrix without a calculator. You may use a calculator to verify your steps, but for full credit your


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USA MA 237 - Linear Algebra Study Guide For The Final

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