Chapter 1 + 2 Flashcards
What is the coefficent matrix?
Augmented matrix without RHS of equation.
T or F. All elementary matrices are invertible.
True.
T or F. Let A be a matrix, and let a and B be EROS, then B(a(A)) = Evb Eva A.
True.
T or F. Applying an inverse ERO to a matrix is the same as applying the ERO to the inverse of the matrix.
True
T or F. If we switch rows, if p is the ERO it is not equal to p^-1.
False.
What is p^-1 if p is an ERO multiplying a row by a scalar?
1 / scalar by the same row.
What is p^-1 if p is an ERO adding a scalar multiple of another row?
p^-1 is subtracting a scalar mutliple from the same row.
T or F. If A is a matrix and p is an ERO than p(A) is row equivalent to A.
True.
T or F. Row equivalent matrices don’t necessarily have the same RREF.
False.
T or F. In REF form, all pivots must be ones.
False.
What is unique between different REF’s?
The number of non-zero rows.
Give some examples of fields.
the rational numbers
the real numbers
the complex numbers
the integers modulo prime p
Give some non-examples of fields.
the integers
the matrices
the quaternions H
the integers modulo non-prime n
T or F. There are three planes in R^3 only two of them intersect how many solutions are there?
None. All three must intersect all three.
Define a field.
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.
Define 7 operation rules for a field.
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
Define a vector space.
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.
Define 8 vector operation laws.
(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
How can you tell whether F +,. or V +,. are being used?
Field + and . are only used for a.b, a+b when both a and b ∈ F.
State Lemma 2.8.
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
T or F. A field is always a vector space of itself.
True
What is the trivial vector space?
{0subV}
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.
True
Define a subspace.
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.
State the Subspace Criterion Theorem.
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.
Define linear-combination.
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
Define the spanning set.
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}
T or F. Span(v1,…,vn) is a subspace of V.
True.
T or F. Span(v1,…,vn) is the smallest subspace of V that contains the vectors v1,…,vn.
True.
Define the spanning set.
If Span(v1,…,vn) = V we say that the vectors v1,…,vn span V.
What do we call V if it has a finite spanning set?
V is finite dimensional
What is a non-finite v.s. called?
Infinite dimensional
Define the rowspace of a matrix A which is an n x m matrix.
R(A) = SpanF(1st row, … , nth row)
Is a subspace of F^m
Define the column space of a matrix A which is a n x m matrix.
C(A) = SpanF(1st column, … , mth column)
Is a subspace of F^n
Define linear independence.
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.
Define linear dependence.
The vectors v1,…,vn ∈ V are linearly dependent (over F) if they are not linearly independent.
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?
det A != 0.
Does it make a difference to linear independence checking if we make v1,…,vn be the rows or the columns of the matrix? Why?
No, det(A) = det(A^T).
We have vectors v1,…,vn ∈ V. These vectors are linearly independent if an only if each vector in Span(v1,…,vn) …
has only one representation as a linear combination of v1,…,vn.
Define the Steinitz Exchange Lemma.
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.
If V is spanned by n vectors, what can we sat about a subset of n+1 vectors in V?
It is linearly dependent.
T or F. If V is finite dimensional its subspace can be infinite dimensional.
False.
Define a basis of V.
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.
Define the standard basis of F^n.
E = (e1 = (1,0,…,0), … , en = (0,0,…,1))
T or F. Every spanning set in a vector space can be reduced to a basis of the vector space.
True.
Define the co-ordinates and give notation.
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)
Define the unique representation property.
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.
T or F. Not every finite-dimensional vector space V has a basis.
False.
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.
True.
State the vector space isomorphism theorem.
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.
If v1,v2,…,vn are n vectors in F^n, state 5 equivalent statements.
- (v1,…,vn) is a basis of F^n.
- The vectors v1,…,vn are linearly independent.
- The vectors v1,…,vn span F^n
- The matrix with these vectors as its rows or columns is invertible.
- The matrix with this vectors as it rows or columns has a REF with n pivots.
T or F. Any two basis of a finite-dimensional v.s. V have the same # elts.
True.
Define the dimension of a v.s. V.
The number of elements in a basis of V.
T or F. dimF {0} := 1.
False, := 0.
State the six dim rules.
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.
If V is a finite dimensional v.s. and U is subspace of V what can we say about dim U and dim V?
dim U <= dim V
dim U = dim V <=> U = V
V is a finite dimensional v.s. every spanning set and every linearly independent set of V that contains dim V vectors is…
A basis of V.
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’.
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’
What is the relation between B and B’ both bases?
New basis equals old basis times A.
B’ = B * A
where A is the CBM from B to B’.
What is the relation between ai and aj’?
Old coefficents equal A times new coefficents.
[] = column vectors
ai = j=1sumn aijaj’ (1 <= j <= n)
[a1,…,an] = A [a1’,….,an’]
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.
A and C are invertible and A^-1 = C and C^-1 = A.
Law for constructing new basis from another using matrices.
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.
Define the null space of a matrix A.
The solution space of the homogeneous system AX = 0subV is called the null space of the matrix A, denoted N(A).
What is the nullity of a matrix A?
The dimension of the null space of A.
T or F. Let B and B’ be two different REFs of A. Then B and B’ have the same number of pivots.
True.
Define row rank of a matrix A.
The dimension of the row space of A.
What is the value of rank(A)?
The number of non-zero rows in any REF of A.
T or F. If a matrix A is row-equivalent to a matrix B its rows have the same linear dependence/independence relation.
True.
Define the column rank of a matrix A.
The dimension of the column space of A.
T or F. Every matrix A has row rank = column rank.
True, defined as rank(A)
Define five rank laws.
- rank(A) = rank(A^T)
- rank(A) <= min(m,n)
- 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.
- rank(A) < n = infinitely many solutions
- rank(A) = n = only trivial solution 0subV
Define the rank-nullity theorem for matrices.
Rank(A) + nullity(A) = n
number of leading variables + number of free variables = n
Maths Tests:
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
T or F. A spanning set of R^3 must have at least 3 elements.
True.
T or F. If a set in R^3 has more than 3 elements it must be linearly DEPENDENT.
True.
Distinguish between span and spanning set of R^3?
A span of R^3 spans a 3-dim subspace of R^3
When this span = R^3 then it is a spanning set.
In a Mn(R) matrix how many entries are on the diagonal or above it?
n(n+1) / 2
What should I do when told to get the row space R(A) that consists of rows of A?
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).
T or F. The span of the empty collection is equal to the trivial vector space. Span() := {0}
True.
What is the basis of a spanning set which only contains the trivial vector 0?
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.
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’?
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
