Lecture Chapter 1 Flashcards
What is a solution set?
The set of all possible solutions of a linear system
What are equivalent systems?
They have the same solution set
What are the possibilities of solutions of a system
either 1, zero or infinitely many solutions
What does consistent mean? inconsistent?
consistent : one or infinitely many solutions
inconsistent : no solution
How many rows and columns has a mxn matrix?
m rows, n columns
three characteristics of echelon form
- all nonzero rows are above any rows of all zeros
- each leading entry is on the right of the one above
- all entries below a leading entry are zeros
three characteristics of reduced echelon form
- echelon form conditions
- the leading entry in each nonzero row is 1
- each leading 1 is the only nonzero in its column
What is a pivot position
leading 1 in the reduced echelon form
What are row equivalent matrices
if row operations can transform one into the other
the two systems are equivalent (same solution set)
Can a matrix be row equivalent to multiple reduced echelon matrices
no, only one
what characterizes a consistent linear system
no [0 0 0 … 0 / b] in the echelon form of the augmented matrix
What is a linear combination
vector y = c1v1 + c2v2 + … + cpvp
linear combination of v1,v2,…,vp with weights c1,c2,…,cp
How do you find the solution of a vector equation x1a1 + x2a2 + … + xnan = b
the equation has the same solution set as the linear system with augmented matrix [a1 a2 … an / b]
What is Span {v1 v2 … vp} ?
The subset of Rn spanned by v1, v2, … , vp
The set of all linear combinations of v1, v2, … , vp (collection of vectors)
What is Ax
Linear combination of the columns of A using the corresponding entries in x as weights
How do you find the solution of Ax=b?
same solution set as
x1a1 + x2a2 + … + xnan = b
[a1 a2 … an b]
What statements are equivalent to ‘for each b in Rm, Ax=b has a solution’
- each b in Rm is a linear combination of the columns of A
- the columns of A span Rm
- A has a pivot position in every row
A(u+v) = Au + Av A(cu) = c(Au)
What is a homogeneous linear system?
Ax=0
Two properties of Ax=0 (homogeneous linear system)
- always has at least one solution : trivial solution : zero vector 0
- if Ax=0 has a nontrivial solution = equation has at least one free variable
parametric vector equation of a line through the origin
x = su
parametric vector equation of a plane through the origin
x = su + tv
parametric vector equation of a line not through the origin
x = su + p
parametric vector equation of a plane not through the origin
x = su + tv + p
When is a set of vectors linearly independent?
When the vector equation x1v1 + x2v2 + … + xnvn = 0
has only the trivial solution
When is a set of vectors linearly dependent? What is the linear dependence relation?
When there exists weights not all zero such that
c1v1 + c2v2 + … + cpvp = 0
(linear dependence relation)
When are the columns of A linearly independent?
If Ax=0 has only the trivial solution
What can you say about a set containing one vector linearly dependent?
It’s the zero vector
What can you say about a set containing two vectors linearly dependent?
The vectors are a multiple of one another
What can you say about a set containing two or more vectors linearly dependent?
At least one is a linear combination of the others
What can you say about a set containing more vectors than there are entries in each vector
The set is linearly dependent
What can you say about a set containing the zero vector
The set is linearly dependent
What can you say about a transformation T from Rn to Rm
- assigns to each x in Rn a T(x) in Rm
- Rn domain
- Rm codomain
- T(x) in Rm is the image of x in Rn
- The set of all images is the range of T
What is T(x)=Ax
matrix transformation
Three properties to show a transformation T is linear
- T(u+v) = Tu + Tv
- T(cu) = cT(u)
- T(0) = 0
What is an identity matrix
1 in the main diagonal, zeros elsewhere
When is T : Rn–>Rm onto Rm?
if each b in Rm is the image of at least one x in Rn
When is T : Rn–>Rm one-to-one?
if each b in Rm is the image of at most one x in Rn
- -> T(x)=0 only has the trivial solution
- -> the columns of A are linearly independent
When does T maps Rn onto Rm?
When the columns of A span Rm
