Math 202                 Study Guide for Hour Quiz #4                     C.S.Davis

Your are responsible for the subject matter in sections 4.1 - 4.7 from the 5th edition of Elementary Linear Algebra by Larson and Edwards. The following topics are the most important. Typical exercises are assigned at the end of each topic. If you need more practice than these, work some more similar exercises.

4.1

1.

Give the definitions for vectors, vector addition and scalar multiplication for vectors in R2, R3 and Rn. Text

2.

Perform vector arithmetic in R2, R3 and Rn. P183#7,11,15,19,23,

3.

Write one vector as a linear combination of other given vectors. P184#35,41

4.2

1.

Give the definition of a vector space. Text

2.

Give lots of examples of vector spaces, and some counter examples of sets and operations that are not vector spaces. For the examples, show that any of the ten properties of a vector space are satisfied. For the counter examples, determine which of the ten properties of a vector space are not satisfied for the operations of addition and scalar multiplication on a given set. P191#1,3,5,7,9,1,13,15,17,18,20,21

3.

Provide reasons for the proofs of the six additional properties of a vector space. P184#48,49,51

4.3

1.

Give the definition of vector subspace. Text

2.

Give examples of subspaces and some counter examples of subsets that are not subspaces. Use the two theorems about necessary and sufficient conditions for a subset to be a subspace and about intersection of subspaces to verify the examples and counterexamples. P200#1-21odd.

4.4

1.

Define "linear combination". Text

2.

Write one given vector as a linear combination of other given vectors if possible. P213#1,3

3.

Define "Spanning Set of a Vector Space", "standard spanning set"and "span(S)". Text

4.

Determine whether a set spans a vector space. P213#5,7,11,13

5.

Define linear independence and linear dependence. Text

6.

Determine whether a set of vectors is linearly independent or linearly dependent. P213#17,21,23,29,33-41odd,43

4.5

1.

Define "basis", "standard basis" and "dimension". Text

2.

Give standard basis for any popular vector space. P224#1,3,4

3.

Determine whether a given set is a basis for a given vector space. If not, explain why not. P224#5-14,15-25odd,29,31,35

4.

Find the dimension of a vector subspace. P224#39-42

5.

Find a basis for a described vector space. P225#49,51,53,57

4.6

1.

Define "row space and column space of a matrix A". Text

2.

Given a matrix A, find a basis for the row space, a basis for the column space and the rank. P239#1,7

3.

Find the span of a set V of vectors. (This is the same as the column space of the matrix A whose columns are the entries of V. I often refer to this matrix as Vcol.) P239#9

4.

Find a basis and dimension of the solution space of Ax=b. (This is the same as the null space and nullity of A, resp.) P239#13,15,17,19,21

5.

Determine whether a nonhomogeneous system Ax=b is consistent and if so, write the solution in the form, x = x_h +x_p, where x_h is a solution of the homogeneous system Ax=0 and x_p is a particular solution of the nonhomogeneous system Ax=b. P240#27

6.

Determine whether a given vector is in the column space of A, and if so, write it as a linear combination of the column vectors of A. P240333,35

7.

Combine practically all of the above in one problem. P241#51

8.

Give equivalent conditions for invertibility of a square matrix.   Text

4.7

1.

Define "coordinate vector of x relative to basis B".   Text

2.

Find the coordinate vector of a given vector relative to a given basis. P253#1,3,7,9,11

3.

Find the transition matrix P from one given basis B' (I use D instead) to another given basis B. Then find the coordinate vector of a given x relative to B and the coordinate vector of x relative D. P253#13,15,17,23,25,27,29,33