MATRIX ALGEBRA & APPLS
MATRIX ALGEBRA & APPLS MA 322
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This 2 page Study Guide was uploaded by Kennith Herman on Friday October 23, 2015. The Study Guide belongs to MA 322 at University of Kentucky taught by A. Sathaye in Fall. Since its upload, it has received 28 views. For similar materials see /class/228134/ma-322-university-of-kentucky in Mathematics (M) at University of Kentucky.
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Date Created: 10/23/15
MA322 Sathaye Know these 0801015 These are the questions that you should know the answers to The knowledge will provide a good preparation for the next text 1 What is a vector space De ne precisely Give examples of set S with operations and such that S is closed under the operations but fails one of the other axioms of the vector space 2 Give ve different examples of vector spaces 3 What is the de nition of a set of independent vectors Give examples of one two and ve independent vectors with justi cation 4 What is meant by the span of a set of vectors What is meant by the statement that a set of vectors is a spanning set of a vector space 5 Give example of a spanning set for a vector space Give example of a set of ve vectors in a vector space V which is not a spanning set for V Justify your claim 6 When is a set of vectors said to be a basis of a vector space 7 Construct an example of a vector space V which has following three examples in it o A set of ve vectors which forms a basis of V o A set of ve vectors which does not form a basis for V These ve vectors should be non zero and none of them should be a multiple of another 0 A set of ve vectors which span V o A set of ve vectors which do not span Vi These ve vectors should be non zero and none of them should be a multiple of another 8 What is meant by the dimension of a vector space 9 Give three distinct examples of an in nite dimensional space 10 What is the de nition of a linear transformation homomorphism of vector spaces 11 Give three examples of linear transformations between in nite dimensional vector spaces 12 Give example of a map between two vectors spaces which is a linear transformation Give example of a map between the same two vector spaces which is not a linear transformation 13 How do you determine if a linear transformation is injective Give an example of the process 14 How do you decide if a linear transformation is surjective Give an example of the process 15 What is meant by the Kernel and the Image of a linear transformation Give at least three examples of linear transformations whose kernels and images are non zero vector spaces 16 What are the two basic ways of describing a subspace of a vector space Answer As a kernel of a linear transformation or as the image of a linear transformation ln 9 this takes on the form of Nul A and Col A Give exaamplesi 17 Given a subspace of the form Nul A how do you write it as Col B for some B 18 Given a subspace of the form Col B how do you write it as Nul A for some A 19 What is the fundamental dimension formula in vector spaces which relates the dimension of a vector space V the dimension of the kernel of a linear transformation of V and the dimension of the image of the same transformation 20 The fundamental dimension formula is related to the number of pivots in the row echelon form of a matrix What is this connection 21 If any independent set of vectors is given in a vector space it can be enlarged to make a basis for the same vector space Give an example of this process where at least two extra vectors are neede i to to N 9 N F to 01 micro mac CADCADCADCADCADCADCADIO OJWHgtCA3MHO CA3 CA3 00 Given any spanning set of a vector space7 it can be trimmed down to a basis Give an example of this process where at least two vectors need to be tr1mme Given a vector space V and an ordered basis B v1 vn what is the de nition of the vector MB What is this vector called Give example of nding such a vector in a nite dimensional space Give a similar example in an in nite dimensional space Given a linear transformation T V A W and ordered bases B of V and C of W how do you nd the ma trix of the transformation with respect to these given bases Answer It is the matrix M whose columns are Tv1cTv2c 7Tvmc where B v1 Um Give examples of such calculations when V is M2309 Same exercise when V Pn for some n Corresponding W can be the same as V or different Given two ordered bases B and C of the same vector space V7 how are they related by a matrix How are the respective coordinates related by the same matrix Answer Write C B Then 11 MMC for all vectors 1 The matrix M is suggestively denoted as PE and its columns are simply the coordinate vectors of members of C with respect to the basis B Give examples of such calculations in 9 Do similar examples in other vectors spaces What is a determinant For which matrices A is det A de ned What are the formulas for a 2 X 2 and a 3 X 3 determinant Which determinants have easy evaluations List as many as possible For example A determinant with a zero column7 a determinant where one row is a sum of other two7 a triangular determinant etc What is the best way of evaluating a large determinant by hand Practice How does a determinant help in determining the rank of a matrix How does one solve linear equations using determinants Cramerls Rule What is an adjugate classical adjoint How is it related to the inverse How is a determinant used to decide if a matrix is invertible What is the relation between det AB and det A and det B What is the connection between a determinant and the area or volume Give concrete uses in 992 and 993 respectively How is a determinant used to write convenient equations for the following geometric objects 0 Equation of a line joining two points in the plane It is a 3 X 3 determinant equated to zero 0 Equation of a plane through three given points It is a 4 X 4 determinant equated to zero How does the general Laplace expansion work Give example of evaluating a determinant using expansion by two I OWS Devise questions of your own to make the number of questions a prime number How many more would be needed
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