Finite Dimensional Vector Spaces

Question 1

Linear combination

Answer 1

A linear combination of vectors v₁,…,vₙ is a vector λ₁v₁ + … + λₙvₙ

Question 2

0 is regarded and a L.C. of …

Answer 2

any set

Question 3

Subspace spanned by A (span A)

Answer 3

is the set of linear combinations of vectors from A

Question 4

proposition 5.4 (span A is a subspace)

Answer 4

if A⊆V the the set of vectors span A is a subspace of V

Question 5

How to find the span of a set of vectors {a,b,c…}

Answer 5

1. put all the vectors in a matrix (as rows)
2. apply row operations to get into echelon form
3. the span of {a,b,c…} is equal to the span of all the non-zero rows

Question 6

How to determine if a vector is part of a spanning set

Answer 6

if a vector can be written as a L.C. of the vectors in the spanning set then it is inside the span

Question 7

How to find a vector that is not in a spanning set

Answer 7

1. take a vector that is in the span

2. change is last element to 0 (or some other value that isn’t a multiple

Question 8

Proposition 5.6. if W⊆V then for A⊆W we have

Answer 8

(1) A ⊆ span(A) ⊆ W
(2) A ⊆ span B so span A ⊆ span B
(3) if x is in span A then span A = span A U {x}

Question 9

linearly dependent

Answer 9

if there is a linear combination λ₁v₁ + … + λₙvₙ = 0 where not all λs are equal to 0

Question 10

linearly independent

Answer 10

not linearly dependent.

Question 11

How to prove a set is linearly independent

Answer 11

1. take an arbitrary L.C. λ₁v₁ + … + λₙvₙ = 0
2. prove all the scalars are equal to 0
   OR
3. put all the vectors in a matrix (as rows)
4. use row operations to get into echelon form
5. if a row can be made as a linear combination of the others then not linearly independent

Question 12

if X is linearly independent OR dependent then any subset of X

Answer 12

remains linearly independent or dependent

Question 13

The empty set is linearly…

Answer 13

independent

Question 14

Proposition 5.11. linear dependence of subsets

Answer 14

if X is linearly dependent then so is every subset of X

Question 15

Theorem 5.12. subset of A that is L.I with the same span.

Answer 15

If A⊆V then there is a linearly independent B⊆A s.t. span A = span B

Question 16

How to find a linearly independent subset B of A with the same spanning set

Answer 16

for each vector in B: if a is linearly independent to current B (start from {}) add a to B. If not linearly independent then discard

Question 17

Theorem 5.14. subset of A that is L.I with the same span (2nd thrm)

Answer 17

If C⊆A⊆V where C is independent, then there is an independent subset B⊆A containing C for which span A = span B

Question 18

basis

Answer 18

a subset B⊆V which is linearly independent s.t. spanB=V

Question 19

The Exchange Lemma

Answer 19

if b∈span(AU{a}) and b∉span(A) then a∈span(AU{b}).

Also if AU{a} is independent then so is AU{b}

Question 20

Theorem 5.20. Sizes of spans.

Answer 20

if A,B⊆V are independent and A⊆spanB then |A|≤|B|

Question 21

span S

Answer 21

is the set of all linear combinations of the vectors in S

Question 22

If C is independent and A is a basis of V then B…

Answer 22

is a basis of V containing C

Question 23

dimension

Answer 23

If B is a basis of V then |B| is the dimension of V

Question 24

Corollary 5.21. size of bases.

Answer 24

if A,B⊆V are bases of V then |A|=|B|