As it was the case with the first one, the best way to prepare for this exam is to go through a summary (see below one or do it yourself!) of the topics discussed so far in class and read/work again the corresponding problems you solved for your homeworks. You might also want to have a look at the problems that I recommended for your individual study. Sometimes one finds later that a solution considered to be "correct" is not so correct anymore; it pays off to check your answers with the ones from the back of the textbook, with those of your colleagues or with me.

In the exam there will be problems testing both your computational and your abstract thinking skills. Let me remind you that only calculators allowing you to perform basic operations may be used. The use during the exam of a calculator that computes the reduced row echelon form or the determinant of a matrix, for instance, will be considered an act of scholastic dishonesty.

I have compiled a summary of the things we did in class after the first exam. While the exam will focus on the new material, you should realize that without a good knowledge of the algorithms for solving linear systems, bringing a matrix into reduced row echelon form, checking linear independence of vectors, finding bases for linear spaces given as the kernel or as the image of a matrix, to name just a few, without these you will not be able to do well on this exam.

The following list covers most of the topics from the book (Section 3.4 through Section 6.3) that we have discussed so far. Notice that the Sections 5.4. and 5.5 are not in this list.

Section 3.4

• compute the coordinates of a vector with repespect to a given basis
• the matrix of a linear transformation T:R^n -> R^n with respect to a given basis and its relation with the standard matrix
• similar matrices

Chapter 4

• examples of linear spaces: matrices, polynomials etc
• checking that some set is a subspace
• linear combinations, span of elements in a linear space
• linear independent and linearly dependent elements in a linear space; how to check that
• basis, coordinates, dimension
• linear transformations, the kernel and the image, the rank-nullity theorem
• isomorphisms and coordinate transforms
• compute the matrix of a linear transformation with respect to some basis; how it changes when we change the basis; the change of basis matrix
• find a basis for the kernel and for the image of a linear transformation

Chapter 5

• orthogonal vectors, the length of a vector
• orthonormal vectors and orthonormal bases for a linear subspace of R^n
• compute the projection of a vector onto a subspace given an orthonormal basis of the latter
• the orthogonal complement of a subspace and its properties
• Pythagoras theorem in R^n and the Cauchy-Schwartz inequality
• the angle between two vectors
• the Gram-Schmidt algorithm for obtaining an orthonormal basis for a subspace
• the QR factorization of matrices
• orthogonal transformations and orthogonal matrices
• the transpose of a matrix

Chapter 6

• the determinant of a square matrix
• the Laplace expansion of the determinant and formulas for the determinant of 2x2 and 3x3 matrices
• how to compute the determinant for special matrices: triangular, partitioned matrices who have a block of zeros
• the effect on the determinant of performing elementary row or column transformations
• use Gaussian elimination to compute the determinant
• check the invertibility of a matrix by looking at its determinant
• the determinant of a product of matrices
• adj(A), the (classical) adjoint of a matrix
• the relation between adj(A) and the inverse of A
• the area of a parallelogram and the volume of a parallelipiped
• compute (up to a sign) det(A) using the QR factorization of A

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