MATH 2331, Linear Algebra
Northeastern University, Fall 2018
Final exam information:
Date: Monday 12/10, 10:30 - 12:30
Location: Behrakis Health Sciences Center 310
There will be a review session on Thursday 12/6, 1:35-2:40, in West Village G106.
The exam is cumulative.
The exam will have 11 problems. It is pretty long, so please do some timed practicing.
No notes, no calculator.
Here are the final exams from Fall 2016 (solutions), Fall 2017 (solutions) and Spring 2017 (solutions).
The emphasis is on applying the concepts and algorithms we have learned, namely:
Algorithms/computations:
Solve linear equations / row reducing a matrix / inverting a matrix.
Given a matrix, determine its rank and nullity, and find a basis for the image and for the kernel.
Given a basis for a subspace, use Gram-Schmidt to convert it into an orthonormal basis. Alternatively, given a matrix, find its QR decomposition.
Calculate the orthogonal projection of a vector onto a subspace. Also, find the matrix of orthogonal projection onto a subspace.
Carry out least-squares approximation, if you're given a dataset and a type of fitting function, or if you're given an inconsistent system of linear equations.
Given a square matrix, find its determinant and decide if it is invertible.
Given a square matrix, find all eigenvalues, and for each eigenvalue, find a basis for the corresponding eigenspace, and determine the algebraic and geometric multiplicities. Determine if the matrix is diagonalizable, and if so, write down its diagonalization in the form D = S^{-1} * A * S or A = S * D * S^{-1}.
Given a diagonalizable square matrix A, find a closed formula for the nth power A^n, and compute the limit of A^n as n goes to infinity.
Given a symmetric matrix, find an orthonormal basis of eigenvectors, and an orthogonal diagonalization D = S^{-1} * A * S or A = S * D * S^{-1}.
Given a quadratic form, convert it to a basis where it has no mixed terms.
Given any matrix, find its singular values and find a singular value decomposition.
Concepts
Given a system of linear equations, determine if it has a unique solution, infinitely many solutions, or no solutions.
Given the rref of a matrix, determine which variables are free variables.
Given a formula for a function from R^n to R^m, determine if it is a linear transformation, and if so, write down its matrix.
Given a description of a linear transformation from R^n to R^m (like "rotation clockwise in R^2 around the origin by 60 degrees, followed by orthogonal projection onto the y axis"), find its matrix.
Be comfortable applying identities like (A*B)^{-1} = B^{-1} * A^{-1} (when A and B are invertible) and (A*B)^T = B^T * A^T.
Understand the meanings of an orthogonal matrix (preserves distances and angles, columns are an orthonormal basis).
Be comfortable using the fact that for an orthogonal matrix Q, the inverse of Q is Q^T.
Be comfortable using the fact that the orthogonal complement to Im(A) is equal to the kernel of A^T.
Be comfortable using the fact that the dot product of v and w is equal to v^T * w (and it is also equal to w^T * v).
Understand how to convert symmetric matrices to quadratic forms and vice versa, and how to determine whether a quadratic form is positive (semi-)definite, negative (semi-)definite, or neither.
Understand the meaning of the singular value decomposition, both in terms of what happens to the orthonormal basis v_1,...v_n, and in terms of the image of the unit sphere.
Starting 11/13, the course TA (Yucheng Liu) will hold recitation sessions for our section, Tuesdays 4pm-5pm in Ryder Hall 159.
Class time and location
Monday, Wednesday, Thursday, 1:35-2:40PM, West Village G108
Office hours:
Me (Lake Hall 463): Mondays 10:35-12:15, Wednesdays 10:30-11:50 (All subject to change soon)
TA: Yucheng Liu (Lake Hall 575): Mondays 11-12, Wednesdays 2:45-4
Information for Midterm Exam:
Date: 11/1, 1:35-2:40
Here is a practice midterm. Solutions are here.
Emphasis will be on Chapter 3 and Sections 5.1,5.2.
I'm not going to ask you to prove, or even state, any theorems. Instead, you need to be comfortable using them. If I ask you to justify your answer, you can say something like "We know v_1,v_2,v_3 form a basis of R^3 since the matrix whose columns are v_1,v_2,v_3 is invertible." (If you were using Theorem 3.3.9.)
Expect the average difficulty of the exam problems to be on the level of the easier Problems (not Exercises) from the homework.
Here is the final exam from Fall 2016. You should be able to do the first five problems.
Material covered:
Chapter 1: Solving linear systems of equations using Gaussian elimination/rref
2.1 Linear transformations: Theorems 2.1.2, 2.1.3 are important. Omit Def 2.1.4
2.2 Geometry of linear transformations: Given a description in words of a linear transformation in the plane, be able to derive its matrix.
2.3 Matrix multiplication: Omit 2.3.9 onwards
2.4 Matrix inverses: Omit Example 7 onwards
3.1 Image and Kernel: Theorems 3.1.3, 3.1.7 are important. Be able to do calculations like example 11.
3.2 Subspaces, linear independence, basis: (Note we used Theorem 3.2.7 as the definition of linear independence.) Theorem 3.2.8 is important. The definition of a basis (3.2.3c) and Theorem 3.2.10 are important.
3.3 Dimensions of subspaces: Theorems 3.3.1, 3.3.2, 3.3.4, 3.3.5, 3.3.6, 3.3.7, 3.3.9 are important. You should be comfortable writing down a basis for the kernel and the image of a given matrix. Be familiar with Summary 3.3.10.
3.4 Coordinates: Use the notation from your notes rather than the book, as the book's notation is confusing. (Use B_old and B_new.) Theorem 3.4.4 is important. So is Example 2.
5.1 Orthogonality: Theorems 5.1.3, 5.1.5, 5.1.6, 5.1.8 are important.
5.2 Gram-Schmidt: Theorem 5.2.1 is important. Be able to turn a basis into an orthonormal basis. (This will be covered on 10/25.)