The second midterm will be on Tuesday, November 19th, from 7:30 to 9:30 pm in Science Center Hall A

You can also stop by to see either Tom or Andy during their office hours (Monday 1:30-3 and 4-5) in their office, room 435 in the Science Center.

Also our Head CA, Mat Sapak, will hold a number of office hours in Loker:
     Sunday night from 8 to 10pm, Monday night from 8 to 10pm and Tuesday from 4 to 6pm

Finally, don't forget to go to your CA's weekly problem session on Monday!!

...and remember  to take advantage of the Math Question Center which meets from Sunday to Thursday from 8 to 10 pm in Loker.

Topics for second midterm:

• Image and Kernel of Linear Transformations:
• Definition of Image and Kernel
• Image = span of column vectors of transformation's matrix
• Kernel = zeros of transformation
• Finding Image and Kernel of a given transformation
• Knowledge of Rank-Nullity relationship for a transformation
• Rank (= dim(Image) plus Nullity (= dim(Kernel) = # of columns of matrix of transformation
• Bases, Span, Linear Dependence/Independence:
• Basic definitions of each concept
• a set of vectors is linearly dependent if and only if there are nontrivial linear relations among them
• the span of a set of vectors is the set of all linear combinations of the set of vectors
• a set of vectors is a basis for a subspace if the subspace is spanned by the set and the vectors are linearly independent
• Ability to find a basis for a given subspace (such as the kernel or image of a linear transformation)
• take spanning set of vectors, form a matrix with vectors as columns, find rref, use pivot columns as a basis
• Study the summary on page 130 for a good summary of chapter 3's concepts
• Subspaces:
• Definition:  subsets of n-dimensional space closed under addition, scalar multiplication (includes zero vector)
• Examples of subspaces
• kernel and image of a linear transformation are both subspaces
• Definition of dimension of a subspace
• number of vectors in a basis for the subspace (uniquely defined, i.e.all bases have same number of vectors)
• Abstract Linear Spaces (Chapter 4)
• Definition - loosely, a set of elements closed under an addition and a scalar multiplication operation, such that the addition and scalar multiplication follow the basic rules as outlined in definition 4.1.1 on page 149 and 150
• Definition of a subspace of a linear space (... a subset that is closed under addition and scalar multiplication)
• Understanding of same concepts as in chapter 3 for general linear spaces:
• ideas of linear independence, span, bases, dimension
• knowledge of linear transformations, kernel and image, rank and nullity (check definition 4.2.1 on page 159)
• Ability to find a basis for a (finite dimensional) linear space given a description of the space
• Definition of an isomorphism - an invertible linear transformation between two linear spaces
• basically identifies the spaces as being "equal" in a sense - same dimension, one to one correspondence of elements
• Topics related to orthogonality (Orthonormal Bases, Orthogonal Complements, etc.)
• Definition of an orthonormal basis
• basis composed of unit length vectors, all orthogonal to one another
• Use of Gram-Schmidt process to find an orthonormal basis for a subspace
• knowledge of QR factorization, ability to find QR factorization while performing Gram-Schmidt
• Definition of an orthogonal complement of a subspace
• all vectors that are orthonal with every vector in the given subspace
• orthogonal complement of a subspace is itself a subspace
• Formula for the orthogonal projection onto a subspace (fact 5.1.6 on page 181)
• Knowledge of Orthogonal transformations
• definition - a linear transformation that preserves length of vectors (or preserves the dot product)
• equivalent properties (summary 5.3.8 on page 207)
• Definition of an orthogonal matrix (the matrix associated to an orthogonal linear transformation)
• products of orthogonal matrices are orthogonal
• Definition of the transpose of a matrix (switches rows and columns)
• definition of symmetric and skew-symmetric matrices (definition 5.3.5 on page 205)
• properties associated to transpose (fact 5.3.9 on page 207)
• orthogonal complement of the Im(A) is the same as the kernel of the transpose of A (fact 5.4.1 on page 212)
• Least Squares Analysis
• Understanding of the normal equation (fact 5.4.6 on page 215)
• Ability to find and interpret results for simple least squares questions (similar to homework problems for section 5.4)
• Note there are several topics in the textbook that we did not cover in class, and which will not be covered on the midterm:
• No coordinates in general linear spaces (section 4.3)
• No inner product spaces (section 5.5)
• Nothing on correlation coefficients (in section 5.1)
• And there are a couple of topics in the textbook that we did in fact cover in class, but which will not be covered on the midterm:
• No coordinate change questions (section 3.4)
• No questions on Cauchy-Schwartz inequality (in section 5.1)

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Old Midterms for practise:

• Note - for this second midterm we have discovered that the midterms from the past several years do not correspond at all to what has been covered in the order we covered the material this semester (there has been a change of editions of the textbook, and our second midterm is about a week earlier than past ones have been).  As a result, there is no point in posting old midterms - the questions on these old midterms are predominantly on eigenvalues and eigenvectors as well as dynamical systems - material we will start to cover right as you take the midterm (in chapter 7), but that will not be included on this midterm this semester.
• As a result, the best source of practice problems is in fact the textbook.  The following is a list of selected problems from each section that will be on the test.  This is not to say that the midterm questions will necessarily look exactly like these questions, but that if you are able to work through these questions and you thoroughly understand what is going on in their solutions, then you should be in good shape for our midterm.
• Practise problems:
• Chapter 3: (solutions)
• Section 3.1 #25, 37, 38, 44, 46, 51
• Section 3.2 #6, 7, 22, 28, 35, 37
• Section 3.3 #8, 22, 35, 36, 37, 38
• Chapter 4: (solutions)
• Section 4.1 #3, 8, 10, 20, 29, 31
• Section 4.2 #1, 3, 5, 22, 26, 34
• Chapter 5: (solutions)
• Section 5.1 #11, 16, 22, 24, 28, 34, 39
• Section 5.2 #4, 8, 18, 22, 32, 39
• Section 5.3 #6, 8, 14, 18, 34, 39
• Section 5.4 #16, 17, 18, 20, 22, 25
• Problem solutions will be posted over the weekend, but be sure to try to do a problem first before looking at the solution!

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Textbook  True/False chapter review problems:

• Another good way to get ready for the midterm is to do the true/false review problems at the end of each chapter that we've covered.  We will post the answers to these true/false problems shortly as well. Note there are no True/False for chapter 4.

Solutions for the True/False questions from Chapter 3 are here.
Solutions for the True/False questions from Chapter 5 are here.
• Note that a few problems (such as #42 in true/false for chapter 5) are on topics we have specifically excluded from the midterm (see list of such topics above), so there are a couple of problems that you shouldn't expect to be able to do - you should be able to figure out which ones these are by checking the list of topics covered/not covered.