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Unformatted text preview: Math 54 Midterm 2 Review You can get a pdf of this review sheet from my website: <http://math.berkeley.edu/ ∼ asmarks> Solutions to the practice problems on this sheet will also eventually be posted at my website. The second midterm will be in lecture, Tuesday Oct 28. The exam will cover material from chapters 46 in Lay. You should expect the questions on the midterm to be very similar in difficulty and style to those on the homeworks. Some strategies for test taking For problems with equal point values, work on the easiest problems first. Don’t get stuck on a single question when there are others you could be doing much more quickly. Read all the questions carefully – you don’t want to lose points for misreading a question. Show all your work. This may increase the amount of partial credit you get, and helps you make fewer mistakes. Explain what you are doing if you use a theorem that is nonobvious. It’s worth taking an extra minute to write up your answers well. Check your work near the end of the exam to make sure that you’ve have answered each question entirely, especially multipart questions. If you have time, check your work for logical and arithmetic mistakes. If you still have time, check your work in any way you can. Solve problems a different way, draw pictures, do special cases, etc. You shouldn’t leave the exam until the last possible moment. Vector Spaces (Chapter 4) • Vector spaces, and subspaces of vector spaces. Linear transformations between vector spaces. Some important examples of subspaces are the span of a set of vectors, the nullspace (also called the kernel) of a linear transformation, the range of a linear transformation, and the column space of a matrix (which is the range of the associated linear transformation). • Linearly independent sets. A set { v 1 ,..., v n } of nonzero vectors is linearly dependent iff some v j is a linear combination of the preceding vectors v 1 ,..., v j 1 . If v j is a linear combination of the preceding vectors, then the set of vectors formed by removing v j still has the same span. • Bases. Be able to find a basis for the nullspace and columnspace of a matrix. Watch out: when you’re finding a basis for the column space of a matrix, don’t use the pivot columns in the row echelon form; you must use the columns in the original matrix corresponding to the pivot columns in the row echelon form. This is because the columnspace of a matrix can be changed by row operations (though the linear dependence relations among the columns are preserved). This is the opposite of what happens for the rowspace; the rowspace of a matrix is preserved by row operations, but the linear dependence relations aren’t . So when finding a basis for the rowspace of a matrix, you need to take the pivot rows from the row echelon form....
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 Summer '09
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 Math, Linear Algebra, basis

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