MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left null space. Linear systems. 3. Subspaces. Linear independence. Span. Basis.. Inverses: left inverse, right inverse, two-sided inverse. 5. Determinants. Methods of computing determinants. Volumes. 6. Orthogonal projections. Least squares. Orthonormal bases. Gram-Schmidt. 7. Matrix of a linear transformation. Composition of linear transformations. Change of basis. 8. Similar matrices. Diagonalizable matrices. Powers of matrices, exponentials, recursions. 9. Eigenvalues. Eigenvectors.. Symmetric matrices. Quadratic forms. The following problems illustrate these topics. Going through all questions in detail is the equivalent of 2 final exams. This is a bit time consuming, but different people need practice with different topics. You don t need to solve every question below, just focus on the topics you feel you need more practice with.. (Null space, column space, row space, left null space, left inverse, right inverse) 3 2 6 3 9 5 A = 2 6 9 5 5. 2 6 2 After carrying out several row operations, we arrive at the matrix 3 3 2 5 7 5.. (i) Give a basis for the null space of A. What is the nullity of A? (ii) Show that the columns c, c 2, c 3, c of A are linearly dependent by exhibiting explicit relations between them. (iii) Give a basis for C(A). What is the rank of A? (iv) Give a basis for the row space of A. (v) What is the dimension of the left null space of A? (vi) Does A admit a left inverse? How about a right inverse?
2. (Linear systems, null space.) Consider a matrix A such that rref A = 2 2 2 and A = 3. (i) Find the set of solutions to the system A x = 3. Is this set of solutions a vector space? (ii) It is known that the second column of A is 2 2 and the fourth column of A is. 2 Find the matrix A. 3. (Orthogonal complements, orthonormal bases, projections.) Consider the subspace V R spanned by the vectors V = span, 3. (i) Find the orthogonal complement V. (ii) Find an orthonormal basis for V. (iii) Calculate the projection of the vector u = 2 7 onto V. (iv) Find the orthogonal projection of the vector u onto V. (v) Write down the matrix of the projection onto V as a product of a matrix and its transpose.. (Projections, left inverse, least squares.) (i) Find the left inverse of A. 2 A = 3 2. 3 (ii) Find the projection matrix onto the column space of A. (iii) Find the projection matrix onto the left null space of A. (iv) Find the least squares solution of the system Ax = 2 3.
5. (Determinants. Invertible matrices.) (i) Calculate the determinant of the matrix A = and the inverse of A. (ii) If AB = BA, can B be invertible? 6. (Similar matrices.) For what values of a, b are the two matrices below similar A =, B = 2. a b 7. (Diagonalizable matrices, exponentials.) Calculate the exponential e ta. 8. (Diagonalizable matrices, recursion.) A = Find the solution of the Fibonacci-like recursion [ ] 3. 2 G n+2 = 3G n+ 2G n, G =, G =. 9. (Symmetric matrices, Gram-Schmidt, quadratic forms.) which has λ = 2 as a repeated eigenvalue. A = 3 3 3 (i) Orthogonally diagonalize A, that is write A = QDQ where Q is an orthogonal matrix. (ii) Show that A is positive definite and find a decomposition A = RR T by any method you wish. (iii) Write the polynomial as sum of three squares.. (Quadratic forms.) f = 3x 2 + 3y 2 + 3z 2 + 2xy + 2yz + 2zx Discuss the definiteness of the following quadratic forms:
(i) Q(x, y) = 2x 2 + 3y 2 xy; (ii) Q(x, y, z) = x 2 + y 2 + z 2 + 6xy + 6xz + 6yz.. (Eigenvalues.) Some of the entries of the following 3 3 matrix have been erased A = 3. However, it is known that the determinant of A equals. (i) Verify that λ = is an eigenvalue for A. (ii) Find the remaining two eigenvalues λ 2 and λ 3. (iii) Is the matrix A diagonalizable? Why or why not? (iv) Find the determinant of the matrix A 2 + 2I. 2. (Eigenvalues, diagonalization.) Let V be subspace of R n of dimension k for < k < n, and let P denote the matrix of the reflection in V, Ref: R n R n. (i) Explain that the eigenvalues of P equal either and, and write down the two corresponding eigenspaces. (ii) Is P diagonalizable? (iii) What is the trace of P? 3. (Symmetric matrices, quadratic forms.) (i) Let A,..., A n be symmetric matrices with non-negative eigenvalues. Prove that det(a +... + A n ). (ii) Consider real square matrices B,..., B n. Using (i), prove that. (Linear systems. Span.) Consider the vectors v = det(b T B +... + B T n B n ). 2, v 2 = 2, v 3 = 2. Find all vectors of the plane {x = x 2 = } in R which belong to span{v, v 2, v 3 }.
5. (Change of basis.) Consider the basis of R 3 given by 2,,. Let T be a linear transformation such that T ( v ) = v + 2 v 2 T ( v 2 ) = v + v 3 T ( v 3 ) = v + v 2 v 3 Find the matrix of the linear transformation T in standard basis. 6. (Geometry of vectors.) Let x, y, z be vectors in R n of lengths, 2, 3 respectively. Suppose that x is parallel and in the same direction as y, while x is perpendicular to z. For what constant c are the vectors x + y + z and x + c y + z perpendicular? 7. (Subspaces.) (Harder) Show that if V and W are two subspaces in R n, then dim(v W ) dim V + dim W n. 8. (Determinants. Diagonalizable matrices. Eigenvalues.) (Harder) Fix a <. Find the volume of the box in Rn spanned by the columns of the matrix... a... A n = a............. a The matrix is tridiagonal: it has s on the diagonal, s right above the diagonal, and a s just below the diagonal.