5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.

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1 Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the solution of the homogeneous system with the same coefficient matrix..) Write the solution you found in # as the sum of a particular solution to the nonhomogeneous system plus the solution to the corresponding homogeneous system. 4.) Explain what it means when we say " v is in the span of { v v v }." Explain what it means when we say " v is not in the span of { v v v }." c.) Explain what it means when we say "{ v v v } spans R." d.) Explain what it means when we say "{ v v v } does not span R." e.) Give an example of vectors u and v in R and another vector w such that w is in the span of {u v }. Be sure to prove that w is in the span of { u v }. f.) Give an example of vectors u and v in R and another vector w such that w is NOT in the span of {u v }. Be sure to prove that w is NOT in the span of {u v }..) For each of the given sets of vectors determine whether or not the set spans R. Give reasons for your answers. {( ) ( )} {( ) ( ) ( )} c.) {( ) ( ) ( )} d.) {( ) ( ) ( )} 6.) Suppose a is a scalar and u and v are vectors in R n. Prove the following. Test # a (u + v ) = a u + a v.) For each set of vectors determine if the set is linear independent and if it spans the given space. Give reasons for your answers.

2 c.).) For each of and (i) Determine if the given function is a linear transformation. Show all necessary work. (ii) For any that are linear transformations give the matrix A such that T(v) = Av. (iii) For any that are linear transformations determine if it is one-to-one. Give a reason. (iv) For any that are linear transformations determine if it is onto. Give a reason..) Prove that a set of two vectors is linearly dependent if and only if one of the vectors is a multiple of the other. 4.) Consider the following matrices. Find each of the following if it exists. If it does not exist explain why not. A + B C + A c.) AB d.) BA e.) D - f.) D g.) C T h.) B -.) Consider the following matrix.

3 A = The matrix when row reduced produces the following matrix. Do the following: If A is a matrix for a linear transformation what is the domain? If A is a matrix for a linear transformation what is the codomain? c.) If A is a matrix for a linear transformation is it one-to-one? Why or why not? d.) If A is a matrix for a linear transformation is it onto? Why or why not? e.) Give a basis for the column space of A. f.) Give a basis for the null space of A. 6.) Suppose T is a linear transformation. Prove that if the only vector x such that Tx = is the zero vector then T is one-to-one. 7.) For each of and prove or disprove whether the given set is a subspace of the given space. a S = 4 a a R V = R v. H = vv V = R v Test #.) Let matrix A be given by A:= By row operations A can be reduced to B below.

4 B:= What is the rank of A? Find a basis for the column space of A. c.) Find a basis for the null space of A. d.) What is the dimension of the null space of A?.) Consider the following basis for R and vector in R. B= {( ) ( ) ( )} v = ( ) Find the coordinate vector for v in terms of B..) Consider S = { x + x + x + x + x + x + } a subset of P. [Hint: Whichever one you do first it can be used to make the other part easy.] Prove or disprove: The set S spans P. Prove or disprove: S is a linearly independent set. 4.) For each of the following matrices find the determinant. 4 use row operations to simplify it and then do it by expanding by some row or column. 44 e 4.7 sinh ) Let T be a linear transformation from the vector space V into the vector space W. Prove that T(V) is a subspace of W. [Note: T(V) is the set of all vectors w in W such that there exists x in V with Tx = w.] 6.) Determine if with the usual addition and scalar multiplication the set of increasing functions is a subspace of the vector space of continuous functions on the real line.

5 7.) Suppose {v... v n } is a basis for a vector space V. If v is in V and v = a v + a v a n v n = b v + b v b n v n. Prove that it must be true that a = b a = b... a n = b n. 8.) Consider the set C of differentiable functions on the real line. Determine whether T(f(x)) = df/dx is a linear transformation on C. Justify your answer. 9.) Determine whether the given set is a subspace of the given space. Justify your answer. a S = a c R V = R. c Test #4 v H = v v v V = R.) Given a scalar c and a vector u in R n prove or disprove cu = c u.) 4 Consider the matrix A =. Write the matrix A - I. Find the characteristic equation. c.) Find the eigenvalues. d.) Is the matrix invertible? (Answer without trying to find the inverse or determinant.) Give a reason for your answer. e.) Is A diagonalizable? Why or why not?.) Consider the matrix A given below. It can be shown that = is an eigenvalue for A. Find the eigenspace corresponding to =. ( ) 4.) Consider the basis B for R and the vector v. {( ) ( ) ( )} ( ) Verify that B is an orthogonal set. Find scalars a b c so that v = a v + b v + c v where v v and v are the vectors in B.

6 B" v c.) Use B to create an orthonormal basis for R. [Hint: this does NOT require Gram-Schmidt].) Use Gram-Schmidt and find an orthonormal basis for the subspace W of R 4 that has the following basis ) Consider the points ( ) ( ) (4) and (). Use the least squares approximation to find the quadratic function f(x) = ax + bx + c that best approximates the data. You may use your calculator but show ALL work ALL matrices used along with some indication of your process. 7.) Consider the following bases for R. Find a single matrix that would convert v B" into v B'. Find the coordinates of v in terms of B'. c.) Find the coordinates of v in terms of the usual basis. 8.) Prove that if A and B are similar matrices and A is invertible that B is invertible. 9.) Let A = 6 u = 6 and v =. Use the definition of eigenvalue and eigenvector to determine if u is an eigenvector of A? If so for what eigenvalue? Use the definition of eigenvalue and eigenvector to determine if v is an eigenvector of A? If so for what eigenvalue? Final Exam For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the solution of the homogeneous system with the same coefficient matrix. B" B

7 .) Write the solution you found in # as the sum of a particular solution to the nonhomogeneous system plus the solution to the corresponding homogeneous system. 4.) Explain what it means when we say " v is in the span of { v v v }." Explain what it means when we say " v is not in the span of { v v v }." c.) Explain what it means when we say "{ v v v } spans R." d.) Explain what it means when we say "{ v v v } does not span R.".) For each of the given sets of vectors determine whether or not the set spans R. Also determine if it is linearly independent. Give reasons for your answers. {( ) ( )} {( ) ( ) ( ) } c.) {( ) ( ) ( ) ( ) } 6.) Suppose a is a scalar and u and v are vectors in R n. Prove the following. a (u + v ) = a u + a v 7.) For each of and (i) Determine if the given function is a linear transformation. Show all necessary work. (ii) For any that are linear transformations give the matrix A such that T(v) = Av. (iii) For any that are linear transformations determine if it is one-to-one. Give a reason. (iv) For any that are linear transformations determine if it is onto. Give a reason.

8 8.) Prove that a set of two vectors is linearly dependent if and only if one of the vectors is a multiple of the other. 9.) Suppose T is a linear transformation. Prove that if the only vector x such that Tx = is the zero vector then T is one-to-one..) Consider the following matrix. A = The matrix when row reduced produces the following matrix. Do the following: If A is a matrix for a linear transformation what is the domain? If A is a matrix for a linear transformation what is the codomain? c.) If A is a matrix for a linear transformation is it one-to-one? Why or why not? d.) If A is a matrix for a linear transformation is it onto? Why or why not? e.) Give a basis for the column space of A. f.) Give a basis for the null space of A..) For each of and prove or disprove whether the given set is a subspace of the given space. a S = 4 a a R V = R. H = v v v v V = R

9 .) Consider S = { x + x + x + x + x + x + } a subset of P. [Hint: Whichever one you do first it can be used to make the other part easy.] Prove or disprove: The set S spans P. Prove or disprove: S is a linearly independent set..) Determine if with the usual addition and scalar multiplication the set of increasing functions is a subspace of the vector space of continuous functions on the real line. 4.) Consider the set C of differentiable functions on the real line. Determine whether T(f(x)) = df/dx is a linear transformation on C. Justify your answer..) Given a scalar c and a vector u in R n prove or disprove cu = c u 6.) Prove that if A and B are similar matrices and A is invertible that B is invertible.