Final EXAM Preparation Sheet

Transcription

1 Final EXAM Preparation Sheet M369 Fall Key concepts The following list contains the main concepts and ideas that we have explored this semester. For each concept, make sure that you remember about it, that you understand it, and that you have a couple examples in mind of that concept in action. 1. Vectors and vector spaces. 2. Subspace of a vector space. 3. Linear independence. 4. Span of a set of vectors. 5. Basis of a vector space. 6. Dimension of a vector space. 7. Linear function. 8. Linear isomorphism. 9. 1:1, onto. 1. Kernel of a linear function. 11. Range of a linear function. 12. Relation between kernel, range and being 1:1 or onto. 13. Rank-Nullity theorem. 14. How a linear function is represented by a matrix when bases are chosen. 15. How the matrix representation changes when bases are changed. 16. The inverse image of a vector via a linear function. 17. The relationship between solving a linear system of equations and looking for the inverse image of a vector via a linear function. 18. The type of solution sets that linear systems can have. 19. The determinant: what it does for us. 2. The determinant: the formal properties it satisfies. 21. How to use such formal properties to compute the determinant. 22. The explicit formula for the determinant in the 2 2 and 3 3 cases.

2 2 True-false questions to review concepts from first and second midterms. 1. A line through the origin is a vector subspace of the cartesian plane. 2. Any line is a vector subspace of the cartesian plane. 3. A parabola through the origin is a vector subspace of the cartesian plane. 4. Five vectors in R 3 cannot be linearly independent. 5. One vector is always linearly independent. 6. Three vectors that span a plane are not linearly independent. 7. Two vectors that span a plane are linearly independent. 8. One vector can span a plane. 9. The span of a set of n linearly independent vectors is a vector subspace of dimension n. 1. A set of linearly independent vectors is a basis for the vector subspace that they span. 11. If V has dimension 4, then any basis for V has four elements. 12. A basis for a vector space V is a subspace of V. 13. Given a set of linearly independent vectors of V that do not span all of V, one can always add some vectors and obtain a basis for V. 14. Given a set of linearly dependent vectors that span V, one can always remove some vectors and obtain a basis for V. 15. If L : V W is a linear function, then the image of a vector subspace of V via L is a vector subspace of W. 16. If L : V W is a linear function, then the inverse image of a vector subspace of W is a vector subspace of V. 17. If L : V W is a linear function and w, it is possible that L 1 (w) is a vector subspace of V. 18. If L : V W is a linear function and {e 1,... e n } is a basis for V, then {L(e 1 ),... L(e n )} is a basis for W. 19. If L : V W is a linear isomorphism and {e 1,... e n } is a basis for V, then {L(e 1 ),... L(e n )} is a basis for W. 2. If L : V W is a linear function and {e 1,... e n } is a basis for V, then {L(e 1 ),... L(e n )} is a basis for Range(L).

3 21. If L : V W is a linear function and {e 1,... e n } is a basis for V, then {L(e 1 ),... L(e n )} span W. 22. If L : V W is an onto linear function and {e 1,... e n } is a basis for V, then {L(e 1 ),... L(e n )} span W. 23. If L : V W is a linear function and {e 1,... e n } is a basis for V, then {L(e 1 ),... L(e n )} span Range(L). 24. If L : V W is a 1 : 1 linear function and {e 1,... e n } is a basis for V, then {L(e 1 ),... L(e n )} is a basis for Range(L). 25. The solution set of a system of linear equations is equal to the inverse image of a vector for some linear function L. 26. Solving a linear system of 3 equations in 7 unknowns corresponds to computing the inverse image of a vector via a linear function L : R 3 R Solving a linear system of 3 equations in 7 unknowns corresponds to computing the inverse image of a vector via a linear function L : R 7 R For a linear system corresponding to a 1 : 1 linear function, there is always a unique solution. 29. For a linear system corresponding to a 1 : 1 linear function, if a solution exists, it is unique. 3. For a linear system corresponding to an onto linear function, a solution always exists. 31. For a linear system corresponding to an onto linear function, the solution set is a translate of the kernel of the linear function. 32. A linear system of equations may have exactly 6 solutions.

4 3 True-False questions about the determinant 1. If A, B are two square matrices, then det(a + B) = det(a) + det(b). 2. If A is a 3 3 square matrix, det(2a) = 2 det(a). 3. The determinant of a diagonal matrix equals the product of the diagonal entries. 4. The determinant of a diagonal matrix equals the sum of the diagonal entries. 5. A linear function represented by a diagonal matrix is invertible if and only if all the diagonal entries are non-zero. 6. Let v 2,..., v n be n 1 vectors in R n. Then is a linear function. 7. det(, v 2,..., v n ) =. 8. det(v 2, v 2,..., v n ) =. det(, v 2,..., v n ) : R n R 9. The determinant of a matrix doesn t change if you scale one of the columns of the matrix. 1. The determinant of a matrix doesn t change if you switch two columns of the matrix. 11. The determinant of a matrix doesn t change if you add to a column of the matrix a linear combination of the other columns

5 det = = = 24 = 6 det = Exercises on the determinant 1. Consider the linear function L : R 4 R 4 defined by L(x, y, z, w) = (x + y, 2x + 3y, 3y + z, x + y + z). Is L an invertible function? 2. Consider the linear function L : R 4 R 4 defined by: L(e 1 ) = e 1 + e 2 + e 3 + e 4 L(e 2 ) = 3e 2 + e 3 + e 4 L(e 3 ) = e 3 + e 4 Is L an invertible function? L(e 4 ) = e 3 e 4

6 3. Determine the dimensions of the Range and the Kernel of the linear function L represented by the following matrix: M L = Determine the dimensions of the Range and the Kernel of the linear function L represented by the following matrix: M L = Decide if the following vectors are linearly independent: v 1 =, v 2 = 3 4, v 3 = Compute the determinant of the matrix M: M = Compute the determinant of the matrix M: M = Compute the determinant of the matrix M: M =