1.7 Key Ideas Flashcards
Linear Dependence is determined by…
the problem Ax = 0. If there is only a Trivial solution it is said o be Linearly Independent. If it has a non-Trivial Solution it is said to be Linearly dependent.
Independent: Ax = 0 ; x = 0 only trivial
Dependent: Ax = 0 ; x = [Solution set] Non-Trivial
When given a set we can determine Linear Dependence by…
Setting up a matrix equivalent to Ax = 0. From here we solve the matrix.
If a free variable shows up, Linearly Dependent.
If an Exact solution arises, Linearly Dependent
If a Trivial solution shows up, Linearly Independent
Linear Independence of Matrix Columns can be determined by the…
equation Ax = 0, thus set up and solve an augment matrix with zeros.
Trivial solution = Independent
Free Variable = Dependent
Exact = Dependent
For ALL Columns making up the matrix. There can not be a dependent column within a a group of independent columns
Linear Dependence of A single vector is determined…
By whether it is a zero vector or not.
Not the Zero Vector = Independent
Zero Vector Dependent
This is because if set up the equation Ax = 0 and the vector is the zero vector you would have two free variables thus ,Linear dependent
Scalar Multiples and Linear dependence…
If a pair of vectors are scalar multiples of one another they will provide an solution containing free variables to Ax = 0 thus, linearly dependent
Linear Dependence within a set of vectors(3)…
If one of the vectors within a group of vectors is a linear combination of another vector within the group the group will be linearly dependent
If the number of vectors outweigh the number of entries within each vector then within a group then the set is Linearly depended, n > m = dependent
If a set contains a zero vector then the set must be Dependent
The following must all be true or false for Ax = b
a. ) The equation Ax = b has a solution
b. ) Each b is a linear combination of the columns of A
c. ) The columns of A span R^m
d. ) A has a pivot position in every row
b is a Linear Combination of Ax when…
There exists a solution for Ax = b
Parametric form is….
x[Matrix]
A Transformation is defined as…
a rule that assigns each vector X in R^n a vector T(x) in R^m
What is the Domain of a transformation defined as?
R^n is called the domain of T.
n = number of Columns in A
What is the Codomain of a transformation defined as?
R^m is called the codomain of T.
m = number of entries in each column of A
Image is…
The vector T(x) in R^m is called the image of x in R^n Under the action of T
The Range of T is defined as…
the set of all images of T(x)
How to check if a Transformation is linear?
It must satisfy the conditions:
a. ) T(u+v) = T(u) + T(v)
b. ) T(cu) = cT(u)
It must preserve the operations of vector addition and scalar multiplication.
