LA Lecture 3 Flashcards
How many components are there in R^n
n components
If π£ and π€ are in a vector space S, every combination ______ must be
in S.
cv + dw
One-point space Z consists of _______
x = 0
y = 3x is in R^___ space.
R^2
Column space of A contains _________.
all combinations of the columns of
A
Ax = b is solvable when _______.
b is in C(A)
Column space is a subspace of ________.
R^m
[4
π] is in ______ space.
R^2
(1, 1,0,1,1) is in _____ space.
R^5
[ 1+ i
1 - i ] is in _____ space.
C^2
What are the requirements for to satisfy to be subspace of a vector space.
- v + w is in subspace
- cv is in subspace
Does the subspace of vector include 0?
yes. th.e subspace of vector include 0
Null space is subspace of _________
R^n
Null space consists all combinations of special solution to _______.
Ax = 0
Row space is subspace of _____.
R^n
For an m by n matrix, the number of pivots variables plus the number of free variables is ______.
n
Counting Theorem: r + (n-r) = n
A square matrix has no free variables. (true/false)
False
An invertible matrix has no free variables.
True
An m by n matrix has no more than n pivot variables.
True
An m by n matrix has no more than m pivot variables.
True
What is the only solution to π΄π₯ = 0 if the columns of π΄ are independent?
π₯ = π
What is the nullspace of a matrix denoted as?
π
What does it mean for vectors to be independent?
The only zero combination π1π£1 + β―+ πππ£π = 0 has all πβπ = 0.
If a matrix has π < π, what can be said about its columns?
The columns are dependent with at least π β π free variables.
When do the vectors π£1,β¦,π£π span a space S?
If S = all combinations of the vβs.
What are the conditions for vectors π£1,β¦,π£π to be a basis for S?
They are independent and they span S.
How is the dimension of a space S defined?
It is the number of vectors in every basis for S.
If π΄ is a 4 by 4 invertible matrix, what can be said about its columns?
Its columns are a basis for π 4.
What is the dimension of π 4?
4
What is the definition of linear independence for the columns of A?
The only solution to π΄π₯ = 0 is π₯ = 0.
If three vectors are not in the same plane, what can be said about their independence?
They are independent.
If three vectors w1, w2, w3 are in the same plane, what can be said about their dependence?
They are dependent.
What is the condition for the columns of A to be independent in terms of rank?
The rank is π = π.
What does it mean if there are n pivots in a matrix?
There are no free variables.
What must be true if n vectors in π π are present?
They must be linearly dependent if π > π.
What defines a set of vectors that spans a space?
Their linear combinations fill the space.
What is the relationship between the columns of a matrix and its column space?
The columns span its column space.
What is the definition of linear independence for a sequence of vectors π£1, π£2,β¦,π£π?
The only combination that gives the zero vector is 0π£1 + 0π£2 + β¦ + 0π£π.
What is the definition of the row space of a matrix?
The subspace of π π spanned by the rows.
How is the row space of A represented?
It is πΆ(π΄^π), the column space of π΄^π.
What are the two properties of a basis for a vector space?
- The basis vectors are linearly independent
- They span the space.
How is the dimension of a vector space defined?
The number of vectors in every basis.
What is the dimension of the whole n by n matrix space?
πΒ²
What is the dimension of the vector space M containing all 2 by 2 matrices?
4
If columns of matrix are dependent, so are the rows.
true/false
False
The column space of 2 by 2 matrix is the same as its row space.
true/false
False
The column space of 2 by 2 matrix has the same dimension as its row space.
true/false
True
The columns of a matrix are a basis for column space.
true/false
False.
What is the dimension of the column space C(A) and the row space C(A^T)?
Both have dimension r (the rank of A).
What is the dimension of the nullspace N(A)?
The dimension is n - r.
What is the dimension of the left nullspace N(A^T)?
The dimension is m - r.
What does elimination produce for the row space and nullspace of A?
It produces bases that are the same as for R.
How does elimination affect the column space and left nullspace?
Elimination often changes them, but their dimensions donβt change.
What is a rank one matrix and its column space basis?
A rank one matrix is A = uv^T; C(A) has basis u.
What is the basis for C(A^T) in a rank one matrix?
C(A^T) has basis v.
If m=n, then the row space of A equals the column space.
true/false
false
The matrices A and βA share the same four subspaces.
true/false
true
If A and B share the same four subspaces, then A is a multiple of B.
true/false
false
