Chapter 4

Question 1

How is a vector space defined?

Answer 1

A ( real ) vector space is a nonemty set V together with two operations, addition and scalar multiplication, that satisfy the following list of axioms.
For all vectors, u, v, w are in V and scalars c and d are real numbers:
1) ( u + v ) is in V (closed under addition)
2) u + v = v + u
3) ( u + v ) + w = u + ( v + w )
4) There is a zero vector in V such that
0 + u = u
5) for each u in V, there is a -u in V such that u + (-u) = 0
6) cu is in V (closed under scalar multiplication)
7) c(u + v) = cu + cv
8) (c + d)u = cu + du
9) (cd)u = c(du)
10) 1*u = u

Question 2

What are the properties of a vector space?

Answer 2

Let V be a vector space. Then the following hold:
i) The zero vector 0 is unique
ii) For each vector u in V, -u is unique
iii) For each vector u in V, 
(0 vector) *u = 0 vector
iv)For each scalar c is a real number, 
c*( 0 vector ) = ( 0 vector )
v) For each vector u in V, -u = -1(u)

Question 3

How is a subspace defined?

Answer 3

A subset H of the vector space B is called a subspace of V if

a) the zero vector is in H
b) for any vectors u, v in H, u+v is also in H
c) for each vector u in H and each scalar c is a real number, c*u is in H

Question 4

Interesting properties of a subspace

Answer 4

Any subset H of a vector space V that satisfies the properties of a subspace is itself a vector space using the same operations as V

Question 5

What are subspaces of any vector space V?

Answer 5

The following are subspaces of any vector space V:

1) { zero vector}
2) V itself
3) Span{ v1, v2, … vn}, where v1, … , vn are vectors in V

Question 6

What are the special subspaces of a vector space V when V is related to a matrix A?

Answer 6

Let A be an m x n matrix. The null space of A, denoted Nul A, is the set of solutions to Ax = 0. The column space of A, denoted Col A, is the set of all linear combinations of the columns of A.
In other words:
Nul A = { x is an element of R^n such that
Ax = 0}
Col A = Span{a1, a2, … , an} where ai is the ith column of A
Col A = { b is an element of R^m such that
b = Ax is consistent}

Question 7

How are linear transformations defined in terms of vector spaces?

Answer 7

Let V, W be (real) vector spaces. A linear transformation from V to W is a function (transformation) T: V -> W such that

i) T(u + v) = T(u) + T(v) for all u, v in V
ii) T(cu) = cT(u) for all u in V and all c is a real number

Question 8

What is the alternate terminology for Nul A and Col A when thinking in terms of linear transformation?

Answer 8

Let T(x) = Ax, where A is an m x n matrix, and T transforms from vector space V to vector space W
Nul A = subset of the domain that maps to the zero vector (pre-image of 0)
Nul A = kernel of T
Nul A is a subspace of V
Col A = subset of codomain that is mapped onto
Col A = range of T
Col A is a subspace of W

Question 9

How is a basis defined?

Answer 9

Let H be a subspace of the vector space V. The set {b1, … , bp} in H is a basis for Hif

i) {b1, … , bp} is linearly independent
ii) H = Span{b1, … , bp}

Question 10

What is the spanning set theorem?

Answer 10

Let H be a subspace of V, where H = Span{v1, … , vp}

a) If some vk is a linear combination of the others, then the set formed by removing vk from {v1, … , vp} still spans H
b) If H does not equal {zero vector}, some subset of {v1, … , vp} is a basis for H

Question 11

Theorem 4.6

Answer 11

The pivot columns of a matrix A form a basis for Col A

Question 12

Theorem 4.7 - The unique representation theorem

Answer 12

Let B = {b1, … , bn} be a basis for a vector space V. Then for each vector x in V, there exists a unique set of scalars c1, c2, … , cn such that:
x = c1b1 + … + cnbn

Question 13

How is coordinate mapping defined?

Answer 13

Suppose B = {b1, b2, b3, … , bn} is a basis for V and a vector x is in V. The coordinates of x relative to the basis B ( or the B-coordinates of x) are the weights c1, c2, … , cn such that x = c1b1 + … + cnbn
If c1, c2, … , cn are the B-coordinates of x, then the vector in R^n:
[x]_B = [c1; c2; … cn]
is the coordinate vector of x (relative to B), or the B-coordinate vector of x. The mapping x -> [x]_B is the coordinate mapping (determined by B)

Question 14

Theorem 4.8

Answer 14

Let B = {b1, … , bn} be a basis for a vector space V. Then the coordinate mapping from x to [x]_B is a one-to-one linear transformation from V onto R^n.

Question 15

Theorem 4.9

Answer 15

Let V be a vector space with basis B = {b1, … , bn}. Then any set in V with more than n elements must be linearly dependent.

Question 16

Theorem 4.10

Answer 16

Let V be a vector space with basis B = {b1, … , bn}. Then every basis of V has n vectors.

Question 17

How is the dimension of a vector space V defined?

Answer 17

The dimension of V, dim V, is the numbers of vectors in a basis for V
If V is the zero space, dim V = o
If V has an infinite basis, V is called infinite-dimensional

Question 18

Theorem 4.11

Answer 18

Let H be a subspace of the finite-dimensional vector space V. Then dim H <= dim V

Question 19

Theorem 4.12 - The basis theorem

Answer 19

Let V be a p-dimensional vector space, p>=1. Then:

i) any linearly independent set of p vectors in V must be a basis for V
ii) any set of p vectors that spans V must be a basis for V

Question 20

Col A

Answer 20

Let A be an m x n matrix. Then

Col A = Span{ columns of A } is a subspace of R^m

Question 21

Row A

Answer 21

Let A be an m x n matrix. Then

Row A = Span{ rows of A } is a subspace of R^n

Question 22

Theorem 4.13

Answer 22

If A and B are row equivalent, then Row A = Row B

Question 23

How is the basis for Row A defined?

Answer 23

The nonzero rows of an RE form of A form a basis for Row A
Note: dim (Row A) = # of nonzero rows in an RE form of A
= # of pivot rows in A
= # of pivot columns in A
= dim (Col A)

Question 24

Definition of rank

Answer 24

The rank of A is the dimension of Col A

Question 25

Theorem 4.14 - The Rank Theorem

Answer 25

Let A be an m x n matrix. Then
rank A = dim (Col A) = dim (Row A) and
rank A + dim (Nul A) = n

Question 26

How to find a basis for Col A

Answer 26

1: Reduce A to RE form U to determine the pivot columns of A
2: B = { pivot columns of A }

Question 27

How to find a basis for Row A

Answer 27

1: Reduce A to RE form U
2: B = { pivot rows of U }

Question 28

How to find a basis Nul A

Answer 28

1: Reduce A to RRE form U
2: Use U to express the solutions to Ax = 0 in parametric-vector form
3: B = { numerical vectors in the parametric-vector form }