Lecture Chapter 4

Question 1

What is a vector space?

Answer 1

nonempty set of vectors such that
u+v is in V
cu is in V

Question 2

What is a subspace of a vector space?

Answer 2

subset H of V with 3 properties

• zero vector is in H
• H is closed under vector addition : u+v is in H
• H is closed under multiplication by scalars : cu is in H

Question 3

If v1,v2,…vp are in a vector space V, what about Span {v1,v2,…,vp}?

Answer 3

Span{v1,v2,…,vp} is a subspace of V

Question 4

What is a null space?

Answer 4

Nul A : set of all solutions of homogeneous equation Ax=0

Question 5

A mxn
Nul A : subspace of Rn
Col A : subspace of Rm

Question 6

What is a column space?

Answer 6

Col A : set of all linear combinations of the columns of A

Col A = Span {a1, a2, … , an}

Question 7

Is Col A all Rm?

Answer 7

only if Ax=b has a solution for every b in Rm

Question 8

What is a linear transformation T from a vector space V into a vector space W

Answer 8

assigns to each x in V a unique T(x) in W such that

• T(u+v) = T(u) + T(v)
• T(cu) = cT(u)

Question 9

What is the kernel of a linear transformation?

Answer 9

the set of all u in V such that T(u)=0

Question 10

What is the range of a linear transformation?

Answer 10

the set of all vectors in W (T(x)) for some x in V

Question 11

When is an indexed set {v1, v2, … , vp} with v1/=0 linearly dependent?

Answer 11

if some vj is a linear combination of the preceding vectors v1, v2,…, vj-1

Question 12

H subspace of vector space V

When is B = {b1, b2,…,bp} a basis for H?

Answer 12

B is a linearly independent set

the subspace spanned by B coincides with H : H=Span{b1,b2,…,bp}

Question 13

What is the spanning set theorem?

Answer 13

S={v1,v2,…,vp} a set in V , H=Span{v1,v2,…,vp}

• if one of the vectors in S is a linear combination of the remaining ones, the set formed by S - this vector still spans H
• if H /= {0}, some subset of S is a basis for H

Question 14

B a basis for V and x in V

What are the B-coordinates of x?

Answer 14

x = c1b1 + c2b2 + … + cnbn

Question 15

What is the coordinate vector of x relative to B

Answer 15

[x]B=(c1,c2,…,cn)

mapping x–>[x]B is the coordinate mapping

Question 16

What is the unique representation theorem

Answer 16

there exists a unique set of scalars such that x=c1b1 + c2b2 + … + cnbn

Question 17

What is the change of coordinates matrix from basis B to standard basis in Rn?

Answer 17

PB = [b1 b2 … bn]
x=PB[x]B
PB^-1 x = [x]B

Question 18

What is a characteristic of the coordinate mapping x–>[x]B

Answer 18

it is a one to one linear transformation from V onto Rn

Question 19

What is isomorphism

Answer 19

one-to-one linear transformation from a vector space onto another

Question 20

Properties about isomorphism

Answer 20

• a vector space V with basis B, containing n vectors is isomorphic to Rn
• Pn and Rn+1 are isomorphic
• n : dimension of vector space V

Question 21

Vector space V, basis B {b1, b2, … , bn}

What if a set in V contains more than n vectors

Answer 21

the set is linearly dependent

Question 22

What if a vector space V has a basis of n vectors?

Answer 22

then every basis of V must have n vectors

Question 23

What about a vector space V spanned by a finite set?

Answer 23

V is finite-dimensional

dim V is the number of vectors in a basis for V

Question 24

What is the dimension of the zero vector space {0}?

Answer 24

zero

Question 25

V finite dimensional
H subspace of V
What can you say?

Answer 25

any linearly independent set in H can be expanded to a basis for H
H is finite-dimensional
dim H <= dim V

Question 26

What is the basis theorem?

Answer 26

V : p-dimensional, p>=1

• any linearly independent set of exactly p elements in V is automatically a basis for V
• any set of exactly p elements that spans V = basis for V

Question 27

How can you find dim Nul A?

Answer 27

number of free variables in Ax=0

Question 28

How can you find dim Col A?

Answer 28

number of pivot columns in A

Question 29

What is the row space of A?

Answer 29

Row A : set of all linear combinations of row vectors
subspace of Rn
Row A = Col A^T

Question 30

If A and B are row equivalent, what can you say about their row space?

Answer 30

they have the same row space

-B in echelon form : nonzero rows of B form a basis for row space of A and B

Question 31

What is the rank of A?

Answer 31

dimension of the column space of A

Question 32

What is the rank theorem?

Answer 32

dim Col A = dim Row A
rank A = number of pivot positions in A
rank A + dim Nul A = n

Question 33

Invertible Matrix Theorem continued

Answer 33

• columns of A form a basis for Rn
• Col A = Rn
• dim Col A = n
• rank A = n
• Nul A = {0}
• dim Nul A = 0