Chapter 1 + 2

Question 1

What is the coefficent matrix?

Answer 1

Augmented matrix without RHS of equation.

Question 2

T or F. All elementary matrices are invertible.

Answer 2

True.

Question 3

T or F. Let A be a matrix, and let a and B be EROS, then B(a(A)) = Evb Eva A.

Answer 3

True.

Question 4

T or F. Applying an inverse ERO to a matrix is the same as applying the ERO to the inverse of the matrix.

Answer 4

True

Question 5

T or F. If we switch rows, if p is the ERO it is not equal to p^-1.

Answer 5

False.

Question 6

What is p^-1 if p is an ERO multiplying a row by a scalar?

Answer 6

1 / scalar by the same row.

Question 7

What is p^-1 if p is an ERO adding a scalar multiple of another row?

Answer 7

p^-1 is subtracting a scalar mutliple from the same row.

Question 8

T or F. If A is a matrix and p is an ERO than p(A) is row equivalent to A.

Answer 8

True.

Question 9

T or F. Row equivalent matrices don’t necessarily have the same RREF.

Answer 9

False.

Question 10

T or F. In REF form, all pivots must be ones.

Answer 10

False.

Question 11

What is unique between different REF’s?

Answer 11

The number of non-zero rows.

Question 12

Give some examples of fields.

Answer 12

the rational numbers
the real numbers
the complex numbers
the integers modulo prime p

Question 13

Give some non-examples of fields.

Answer 13

the integers
the matrices
the quaternions H
the integers modulo non-prime n

Question 14

T or F. There are three planes in R^3 only two of them intersect how many solutions are there?

Answer 14

None. All three must intersect all three.

Question 15

Define a field.

Answer 15

A field is a triple (F, +, .)
where F: non-empty set, + and . are binary ops. Contains 0subF and 1subF such that 0subF != 1subF.
Also 7 operations rules hold true.

Question 16

Define 7 operation rules for a field.

Answer 16

For all a,b,c ∈ F
(i) a + b ∈ F a.b ∈ F
(ii) Commutative + and .
(iii) Associative + and .
(iv) Distributed . over +
(v) a + 0subF = a and 1subF.a = a
(vi) For all a ∈ F, there is elt denoted -a ∈ F s.t. a + (-a) = 0subF
(vii) For all a ∈ F \ {0} there is an elt a^-1 s.t. a . a^-1 = 1subF

Question 17

Define a vector space.

Answer 17

Let F be a field. A vector space over F is a non-empty set V, with two operations:
x + y ∈ V -> x + y ∈ V
c ∈ F, x ∈ V -> c . x ∈ V
also 8 vector operation laws hold true.

Question 18

Define 8 vector operation laws.

Answer 18

(V1) x + y = y + x
(V2) x + (y + z) = (x + y) + z
(V3) There is a unique vector 0subV ∈ V such that x + 0subV = x
(V4) For every x ∈ V there is a unique vector -x such that x + (-x) = 0subV
(V5) 1subF.x = x
(V6) (cd)x = c(dx)
(V7) c(x + y) = cx + cy
(V8) (c + d)x = cx + dx

Question 19

How can you tell whether F +,. or V +,. are being used?

Answer 19

Field + and . are only used for a.b, a+b when both a and b ∈ F.

Question 20

State Lemma 2.8.

Answer 20

Let V be a vector space with scalar field F.
(1) Given x,z ∈ V there is a unique y ∈ V s.t. x + y = z
(2) λ.0subV = 0subV for all λ ∈ F
(3) 0subF.v = 0subV for all v ∈ V
(4) (-1)v = -v for all v ∈ V
(5) If v != 0subV then λ.v = 0subV implies λ = 0

Question 21

T or F. A field is always a vector space of itself.

Answer 21

True

Question 22

What is the trivial vector space?

Answer 22

{0subV}

Question 23

T or F. A is not a vector space of the field is equivalent to the statement the field does not contain the zero vector.

Answer 23

True

Question 24

Define a subspace.

Answer 24

Let V be a F v.s. W ⊆ V.
W is a subspace of V if W itself is an F v.s. with respect to the ops + and s.m. that W inherits from V.

Question 25

State the Subspace Criterion Theorem.

Answer 25

V is an F v.s., then a subset W of V is a subspace. if an only if (s1) the zero vector 0subV belongs to W (s2) For all u,v ∈ W and for all m,k ∈ F, we have mu + kv ∈ W.

Question 26

Define linear-combination.

Answer 26

A vector v is called an F linear combination of vectors v1,v2,...,vn if there are scalars c1,c2,...,cn ∈ F s.t. v = c1v1 + c2v2 + ... + cnvn

Question 27

Define the spanning set.

Answer 27

Let v1,...,vn be nonzero vectors in V. The set of all F-linear combinations of v1,..,vn is called the F-linear span of v1,...,vn, denoted SpanF(v1,...,vn). Thus: SpanF(v1,...,vn) := {a1v1 + ... + anvn | a1,...,an ∈ F}

Question 28

T or F. Span(v1,...,vn) is a subspace of V.

Answer 28

True.

Question 29

T or F. Span(v1,...,vn) is the smallest subspace of V that contains the vectors v1,...,vn.

Answer 29

True.

Question 30

Define the spanning set.

Answer 30

If Span(v1,...,vn) = V we say that the vectors v1,...,vn span V.

Question 31

What do we call V if it has a finite spanning set?

Answer 31

V is finite dimensional

Question 32

What is a non-finite v.s. called?

Answer 32

Infinite dimensional

Question 33

Define the rowspace of a matrix A which is an n x m matrix.

Answer 33

R(A) = SpanF(1st row, ... , nth row) Is a subspace of F^m

Question 34

Define the column space of a matrix A which is a n x m matrix.

Answer 34

C(A) = SpanF(1st column, ... , mth column) Is a subspace of F^n

Question 35

Define linear independence.

Answer 35

The vectors v1,...,vn ∈ V are linearly independent (over F) if the only choice of scalars a1,...,an ∈ F that makes a1v1 + ... + anvn = 0subV is a1 = ... = an = 0.

Question 36

Define linear dependence.

Answer 36

The vectors v1,...,vn ∈ V are linearly dependent (over F) if they are not linearly independent.

Question 37

Let v1,...,vn be n vectors in F^n. Let A ∈ Mn(F) be the matrix whose columns are A = (v1^T, v2^T ... Vn^T). How can we know if v1,...,vn are linearly independent?

Answer 37

det A != 0.

Question 38

Does it make a difference to linear independence checking if we make v1,...,vn be the rows or the columns of the matrix? Why?

Answer 38

No, det(A) = det(A^T).

Question 39

We have vectors v1,...,vn ∈ V. These vectors are linearly independent if an only if each vector in Span(v1,...,vn) ...

Answer 39

has only one representation as a linear combination of v1,...,vn.

Question 40

Define the Steinitz Exchange Lemma.

Answer 40

Let V be a finite dimensional F v.s. Then # elts in any linearly independent subset of V is <= # elts in any spanning set of V.

Question 41

If V is spanned by n vectors, what can we sat about a subset of n+1 vectors in V?

Answer 41

It is linearly dependent.

Question 42

T or F. If V is finite dimensional its subspace can be infinite dimensional.

Answer 42

False.

Question 43

Define a basis of V.

Answer 43

A basis of V is a tuple of vectors in V that is linearly independent and that spans V (is a spanning set). The plural is bases.

Question 44

Define the standard basis of F^n.

Answer 44

E = (e1 = (1,0,...,0), ... , en = (0,0,...,1))

Question 45

T or F. Every spanning set in a vector space can be reduced to a basis of the vector space.

Answer 45

True.

Question 46

Define the co-ordinates and give notation.

Answer 46

If B = (v1,...,vn) is a basis of V and v = a1v1 + ... + anvn, then we call the tuple of unique coefficients (a1,...,an) ∈ F^n the co-ordinates of v with respect to the basis B. [[v]]subB = (a1,...,an)

Question 47

Define the unique representation property.

Answer 47

A tuple (v1,...,vn) of vectors in V is a basis of V if and only if every v ∈ V can be written uniquely in the form v = a1v1 + ... + anvn where a1,...,an ∈ F.

Question 48

T or F. Not every finite-dimensional vector space V has a basis.

Answer 48

False.

Question 49

T or F. Every linearly independent set of vectors in a finite dimensional vector space can be extended to a basis of the vector space & Every spanning set of V can be reduced to a basis of V.

Answer 49

True.

Question 50

State the vector space isomorphism theorem.

Answer 50

Let V be a finite dimensional F v.s. with basis B = (v1,...,vn). There is a mapping: v = a1v1 + ... + anvn ∈ V <--> (a1,...,an) ∈ F^n. We say that V and F^n are isomorphic.

Question 51

If v1,v2,...,vn are n vectors in F^n, state 5 equivalent statements.

Answer 51

1. (v1,...,vn) is a basis of F^n. 2. The vectors v1,...,vn are linearly independent. 3. The vectors v1,...,vn span F^n 4. The matrix with these vectors as its rows or columns is invertible. 5. The matrix with this vectors as it rows or columns has a REF with n pivots.

Question 52

T or F. Any two basis of a finite-dimensional v.s. V have the same # elts.

Answer 52

True.

Question 53

Define the dimension of a v.s. V.

Answer 53

The number of elements in a basis of V.

Question 54

T or F. dimF {0} := 1.

Answer 54

False, := 0.

Question 55

State the six dim rules.

Answer 55

dimsubF F^n = n dimsubF M(m,n) F = mn dimsubR R3[x] = 4 dimsubR C = 2 dimsubC C = 1 dimsubR R[x] = infinity/undefined.

Question 56

If V is a finite dimensional v.s. and U is subspace of V what can we say about dim U and dim V?

Answer 56

dim U <= dim V dim U = dim V <=> U = V

Question 57

V is a finite dimensional v.s. every spanning set and every linearly independent set of V that contains dim V vectors is...

Answer 57

A basis of V.

Question 58

Field: F, V finite dimensional F v.s. B = (v1,...,vn) , B' = (v1',...,vn') two F bases of V. v = a1v1 + ... + anvn. v = a1'v1' + ... + an'vn'. For all v ∈ V. Define the change of bases matrix from B to B'.

Answer 58

vj' = i=1sumn(aijvi) (1 <= j <= n) The n x n matrix A = aij is called the change of basis matrix from B to B'

Question 59

What is the relation between B and B' both bases?

Answer 59

New basis equals old basis times A. B' = B * A where A is the CBM from B to B'.

Question 60

What is the relation between ai and aj'?

Answer 60

Old coefficents equal A times new coefficents. [] = column vectors ai = j=1sumn aijaj' (1 <= j <= n) [a1,...,an] = A [a1',....,an']

Question 61

Let A be the CBM from B to B' and let C be the CBM from B' to B. What is the relation between A and C.

Answer 61

A and C are invertible and A^-1 = C and C^-1 = A.

Question 62

Law for constructing new basis from another using matrices.

Answer 62

Let (v1,...,vn) be a basis of V. vj' = i=1sumn aij vi where A is a matrix aij. Then (v1',...,vn') is a basis of V if and only if A is invertible.

Question 63

Define the null space of a matrix A.

Answer 63

The solution space of the homogeneous system AX = 0subV is called the null space of the matrix A, denoted N(A).

Question 64

What is the nullity of a matrix A?

Answer 64

The dimension of the null space of A.

Question 65

T or F. Let B and B' be two different REFs of A. Then B and B' have the same number of pivots.

Answer 65

True.

Question 66

Define row rank of a matrix A.

Answer 66

The dimension of the row space of A.

Question 67

What is the value of rank(A)?

Answer 67

The number of non-zero rows in any REF of A.

Question 68

T or F. If a matrix A is row-equivalent to a matrix B its rows have the same linear dependence/independence relation.

Answer 68

True.

Question 69

Define the column rank of a matrix A.

Answer 69

The dimension of the column space of A.

Question 70

T or F. Every matrix A has row rank = column rank.

Answer 70

True, defined as rank(A)

Question 71

Define five rank laws.

Answer 71

1. rank(A) = rank(A^T) 2. rank(A) <= min(m,n) 3. Assume m = n. Then rank(A) = n if and only if A is invertible if and only if the rows of A are linearly independent if an only if the columns of A are linearly independent. 4. rank(A) < n = infinitely many solutions 5. rank(A) = n = only trivial solution 0subV

Question 72

Define the rank-nullity theorem for matrices.

Answer 72

Rank(A) + nullity(A) = n number of leading variables + number of free variables = n

Question 73

Maths Tests:

Answer 73

If det != 0 linear independent If at least n vectors to span a space of dimension n than is spanning set. Basis is linearly independent spanning set Dimension of R-linear span is the number of linearly independent vectors

Question 74

T or F. A spanning set of R^3 must have at least 3 elements.

Answer 74

True.

Question 75

T or F. If a set in R^3 has more than 3 elements it must be linearly DEPENDENT.

Answer 75

True.

Question 76

Distinguish between span and spanning set of R^3?

Answer 76

A span of R^3 spans a 3-dim subspace of R^3 When this span = R^3 then it is a spanning set.

Question 77

In a Mn(R) matrix how many entries are on the diagonal or above it?

Answer 77

n(n+1) / 2

Question 78

What should I do when told to get the row space R(A) that consists of rows of A?

Answer 78

Take A transpose reduce to REF, the columns of A^T that contain pivots form a basis of C(A^T) and hence the R(A).

Question 79

T or F. The span of the empty collection is equal to the trivial vector space. Span() := {0}

Answer 79

True.

Question 80

What is the basis of a spanning set which only contains the trivial vector 0?

Answer 80

Given by empty set = Null. E.g. When N(A) = ( (0,0,0)^T ). Nullity is 0, so we define basis of N(A) as null.

Question 81

Let B = {(1,0)^T,(0,1)^T} and B' = {(2,1)^T,(-1,1)^T}. Express a vector v that's from B in the new basis B'?

Answer 81

Express B' as a matrix P = [2 -1 1 1] and find the inverse. This inverse multiped by v on the right is the answer. We write it as such [[v]]sub B' = P^-1 * v