The best way to prepare for the 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 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. The list covers most of the topics from the book (Section 1.1 through Section 3.3) that we have discussed so far.

Chapter 1

• elementary row operations
• compute the reduced row echelon form of a matrix A ( rref(A) )
• solve linear systems by Gauss-Jordan elimination
• determine the number of solutions of a linear system
• compute the rank of a matrix
• linear combinations of vectors
• compute the product Ax between a matrix A and a vector x (when possible)
• write a linear system in matrix form
• write the solution of a linear system in parametric form (i.e. as vectors)

Chapter 2

• linear transformations: Definition 2.1.1. and equivalent definition in Fact 2.1.3
• interpretation of the columns of the matrix of a linear transformation (Fact 2.1.2)
• check that a given map is a linear transformation
• examples from geometry:
  • projections onto a line in the plane
  • projections onto a line/plane in the 3 dim'l space
  • reflection over a line in the plane
  • reflection over a line in the 3 dim'l space
  • scalings
  • rotations
• invertible matrices:
  • determine if a matrix A is invertible (Fact 2.3.3) and if so, compute its inverse (Fact 2.3.5)
  • Notice that for 2x2 matrices there is the simpler formula from Fact 2.3.6. Feel free to use is dirrectly without making use of the general algorithm.
• the product of two matrices A.B
  • interpretation of the column vectors of A.B
  • the product of two matrices corresponds to composition of linear transformations
  • multiply 2 given matrices
  • compute the inverse of (A.B) when A and B are invertible matrices (Fact 2.4.12)

Chapter 3

• linear subspaces: definition, examples, recognition
• the image and kernel of a linear transformation / of a matrix
• linearly independent vectors and linearly dependent vectors
• basis of a subspace
• given a set of vectors decide if they are linearly independent or not
• decide if a vector is a linear combination of a given set of vectors
• decide if the kernel of a matrix is zero vector (Fact 3.2.9)
• basis and unique representation (Fact 3.2.10)
• dimension of a linear subspace and ways of computing it (Fact 3.3.4)
• algorithms for constructing bases for Im(A) and Ker(A)
• the Rank-Nullity Theorem (Fact 3.3.7)

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