Math 3410 Midterm 1

Front 1

Vector Spaces

Back 1

A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold:

- commutativity

- associativity

- additive identity

- additive inverse

- multiplicative identity

- distributive properties

Hint 1

6 properties

Front 2

Subspaces

Back 2

A subset U of V is called a subspace of V if U is also a vector space

Front 3

Commutativity:

Back 3

u+v = v+u for all u,v∈V

Front 4

Associativity

Back 4

(u+v)+w = u+(v+w) and (ab)v = a(bv) for all u,v,w∈V

Front 5

Additive Identity

Back 5

there exists an element 0∈V such that v+w=0 for all v∈V

Front 6

Additive Inverse

Back 6

For ever v∈V, there exist w∈V such that v+w=0

Front 7

Multiplicative Identity

Back 7

for all v∈V, 1·v=v

Front 8

Distributive Property

Back 8

a(u+v)= au+av and (a+b)v= av+bv for a; ab,∈F and all u,v∈V

Front 9

Conditions for a subspace

Back 9

A subset U of V is a subspace if and only if U satisfies the following three conditions:

i) additive identity: 0∈U

ii) closed under addition: u,w∈U implies u+w∈U

iii) closed under scalar multiplication: a∈F and u∈U implies au∈U

Front 10

Intersection

Back 10

the intersection U∩W of a vector space V is such that u∈U∩W implies u∈U and u∈W.

Front 11

Sum of subspaces

Back 11

the sum of two subspaces U and W of a vector space V is the set U+W={u+w|u∈U and w∈W}

- V⊆U+W: for any v∈V, there exists some u∈U and w∈W such that u+w=v

- U+W⊆V: U and W are subsets of V, so u∈U and w∈W are elements of V

Front 12

Direct Sum

Back 12

We say that a vector space V is the direct sum of subspaces U and W, and write V=U⊕W, if V=U+W and for any v∈V there exist unique vectors u∈U and w∈W such that v=u+w

Front 13

Conditions of a Direct Sum

Back 13

The following are equivalent:

a) U+W is a direct sum

b) If 0=u+w, with u∈U and w∈W, then u=0 and w=0

c) U∩W = {0}

Front 14

Linear combination

Back 14

a linear combination of vectors v₁,...,vm in a vector space V is any vector of the form

v=c₁v₁+c₂v₂+...+cmvm

Front 15

Span:

Back 15

the span of a set of vectors {v₁,...,vm} in a vector space V is any vector space V is the set of all possible linear combinations of those vectors

span{v₁,...,vm} = {a₁v₁+...+amvm|a₁,...,am∈F}

If we say V=span{v₁,...,vm}, we are saying that any vector vector v∈V can be written as a linear combination of the vectors v₁,...,vm. So, V=span{v₁,...,vm} if and only if for every v∈V, there exist a₁,...am∈F such that

v=a₁v₁+...+amvm

Front 16

Finite-dimensional

Back 16

A vector space V is finite dimensional if there exists a finite spanning set for V; that us, if V=span{v₁,...,vm} for some finite set of vectors v₁,...,vm, then V is finite dimensional

Front 17

Linear independence

Back 17

A vector space V is linearly independent of the only choice a₁,...,am∈F that makes a₁v₁,...,amvm =0 is a₁,...,am = 0 (unique representation- vectors in a set are not linear combinations of each other)

Front 18

Linear dependence

Back 18

A set of vectors v₁,...,vm ∈V is linearly dependent of there exist some a₁,...,am ∈F , not all 0 such that a₁v₁,...,amvm =0

Front 19

Basis

Back 19

A basis of V is a set of vectors in V that is linearly independent and spans V.

Front 20

Dimension

Back 20

The dimension of a finite-dimensional vector space is the length of any basis of the vector space. Denoted dim V.

Front 21

Linear transformation

Back 21

a function T:V→W from a vector space V to a vector space W is a linear transformation if

- T(u+v)=T(u)+T(v) for any u,v∈V (additivity)

- T(cv)=cTv for any c∈F, v∈V (homogeneity)

Front 22

Null space

Back 22

The null space of T, denoted Null T, is the subset of V consisting of those vectors that T maps to 0.

The null space of a linear transformation T:V→W is the set null T ={v∈V|Tv=0)⊆V.

Front 23

Range

Back 23

The range of a linear transformation T is the set range T={Tv|v∈V}⊆W.

The range T is a subspace of W consisting of those vectors of the form Tv for some v∈V.

Front 24

Injective

Back 24

A function T:V→W is called injective if Tu=Tv implies u=v

(one-to-one)

Front 25

Surjective

Back 25

A function T:V→W is called surjective if its range equals W.

Front 26

Subspace Test

Back 26

a subset U⊆V is a subspace provided that 0∈U, and U is closed under both addition and scalar multiplication

Front 27

The intersection of two subspaces is a subsapce

Back 27

If U⊆V and W⊆V are subspaces, then U∩W⊆V is a subspace.

Front 28

The sum of two subspaces is a subspace

Back 28

If U⊆V and W⊆V are subspace, then U+W⊆V is a subsapce

Front 29

Conditions of a direct sum

Back 29

A sum U+W=V is a direct sum if U∩W={0} or if u+w=0 when u∈U = 0 = w∈W

Front 30

The span of any set of vectors is a subspace.

Back 30

The span, span{v₁,...,vm} of any set of vectors v₁,...,vm ∈V is a subspace of V

Front 31

Length of linearly independent list ≤ length of spanning list

Back 31

In a finite dimensional vector space, the length of every linearly independent list of vectors is less tha or equal to the length of every spanning list of vectors

Front 32

Criterion for basis

Back 32

If B={v₁,...,vm} is a basis for a vector space V, then every v∈V can be written uniquely as a linearly combination

v=a₁v₁+...+amvm

Front 33

Any spanning set contains a basis

Back 33

given A={v₁,...,vm}, if spanA = V, then there is some basis B of V with B⊆A.

every spanning set in a vector space can be reduced to a basis of the vector space

Front 34

Any linearly independent set can be extended to a basis

Back 34

given a linearly independent set A={v₁,...,vm}, we can find vectors w₁,...,wk not in the span of A such that B={v₁,...,vm,w₁,...,wk } is a basis of V

Front 35

Every subspace of is part of a direct sum equal to V

Back 35

Suppose V is finite-dimensional and U is a subspace of V. Then there is a subspace W of V such that V=U⊕W.

Front 36

Basis length does not depend on basis

Back 36

The number of vectors in any basis for a finite-dimensional vector space V is the same.

Front 37

Dimension of a subspace

Back 37

If V is finite-dimensional and U is a subspace of V, then dim U≤ dim V

Front 38

Linear independent list of the right length is a basis

Back 38

If dimV=n, then any linear independent set of vectors with dim=n is a basis for V.

Front 39

Spanning set of the right length is a basis

Back 39

If dim V=n, then any spanning set of vectors with dim=n is a basis for V

Front 40

Dimension of a sum

Back 40

If U₁ and U₂ are subspaces of a finite-dimensional vector space V, then

dim (U₁+U₂) = dim U₁ + dim U₂ - dim (U₁∩U₂)

Front 41

Linear maps take 0 to 0

Back 41

Suppose T is a linear map from V to W. Then T(0)=0

Front 42

The null space is a subspace

Back 42

Suppose T:V→W. Then null T is a subspace of V

Front 43

Injectivity is equivalent to null space = {0}

Back 43

T:V→W is injective if and only if null T = {0}

Front 44

The range is a subspace

Back 44

the range of a linear transformation T:V→W is a subspace of W.

Front 45

Surjectivity is equivalent to range T = W

Back 45

T:V→W is surjective if range T = W

Front 46

Fundamental Theorem of Linear Maps

Back 46

Suppose V is finite-dimensional and T:V→W. Then range T is finite dimensional and

dim V = dim null T + dim range T