LINEAR ALGEBRA

Syllabus

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Announcements

We covered the definition of "Vector Space" and gave several examples of vector spaces. We also proved propositions 1.1,1.2 and their corollaries.

Suggested excercises for Section 1.2.: 8,9,12,13,17,18,19 and 21.

Jan 14.

We covered the definitions of "Subspace" and "Field" in p.510.

Suggested problems for 1.3: 1.

Jan 15.

Suggested problems for 1.3: 5,8,9,19,20,23.

Jan 16.

Suggested problems for 1.4:3(d),(f), 4(a),(e), 8, 10,11,12,13.

Jan 18.

Suggested problems for 1.5:12,15,17.

Jan 22.

We started SEction 1.6 and covered the definition of "basis" and proved Theorem 1.7.

Exam 1 is going to be on next Wednesday, Jan.30th. Here is the list of things one should prepare:

1.The definition of "Vector Space" and be able to use the definition to check whether a give set is a vector space or not.

Verify whether a set is a subspace or not.

Understand the concept of linear (in)dependency and be able to see if a set is linear independent or not.

Understand the concept of basis and be able to verify whether a given set of vectors forms a basis or not.

The problems are computational in nature although some have little proofs in them, these proofs are routin and simple.

Jan 23.

Suggested problems for 1.6:10,11,12,13,14,16,25.

Jan 29.

Review for Exam 1.

Jan 30.

Exam 1.

Feb.1.

We covered Theorems 2.1 and 2.2.

Suggested exercises for 2.1: 6,10,11,12,13,14,16,17,18,19.

Feb 4.

We proved Theorems 2.3,2.4 and 2.5.

Feb.5

Section 2.1 is finished.

Feb. 8

The problems for take-home exam are as follows:

1. # 13, p.52.

2. # 11, p.52.

3. # 16, p.53.

4. For what value(s) of a will the vectors (1,2,3), (2,-1,4) and (3,a,4) be linearly dependent?

5.Let P be the vector space of all polynomials of degrees less than or equal to 3 and let S be the subset of P with an additional property that p(0) = 0 for every element p(x) in S.

(a)Show that S is a subspace of P.

(b)What is the dimension of S?

(c)Find a basis for S.

Due Monday Feb. 18.

Since you have a week to complete this test, you must present your work neatly and solutions must be well written and do not leave out any crucial details.

I will take the average of the two exams for the exam 1. It it optional, however, if you are satisfied with the score of your exam 1.

Feb.11

Suggested problems for 2.2:5,8,9,12.15.

Feb 13

Section 2.3 is covered up to Theorem 2.12.

Feb.18

Section 2.3 is finished.

Suggested problems for 2.3:1,10,11,12.

Feb.19

Suggested problems for 2.4:2,3,4,5,7,12,13,14.

Feb 20

Section 2.4 is finished.

Exam 2 covers Sections 1.6,2.1,2.2,2.3,2.4.

To prepare the exam you should

be able to verify whether a set is a basis of a vector space and be able to compute the dimension of a given vector space.

understand the Dimension Theorem and its implications, that is using it to show whether a given linear transformation is one to one, onto or both.

know the matrix representation of a linear transformation with respect to the given bases.