Term Test 1 Flashcards
Systems of linear equations can have what 3 types of solutions?
- No solution
- One solution
- Infinitely many solutions
What is an inconsistent vs consistent matrix?
Inconsistent: No solution (row of zeroes = #)
Consistent: one or infinitely many solutions
What is row echelon form?
Rows of zeroes are at the bottom
The leading entry of each row is to the right of the row above’s leading entry
What is Reduced Row Echelon form?
REF
All leading entries are 1
Each leading entry is the only non zero in it’s column
What are basic and free variables?
Basic variables are pivots
Free variables are non pivots and can be anything
What is a linear combination?
It’s the sum of the products of constants and vectors (y=c1v1+c2v2+…)
What is span?
The collection of all possible linear combinations of vectors AKA the set of vectors generated by v1,v2…
What is the matrix equation?
Ax=b
When can you multiply two matrices?
When the first matrix has the same number of columns as the second matrices rows
What is a homogenous system?
When ax=o (it can never be inconsistent)
What is parametric vector form?
When the answer is written as a sum of constants multiplied by column vectors
What is a non homogenous system?
When ax=b (but still related to homogenous)
What is the trivial and non trivial solution to homogenous systems?
Trivial: x=o
Non trivial: anything else that is a solution
What is linear independence?
If the vector equation has only the trivial solution
What is linear dependence?
If the vector equation has a non trivial solution (free variable makes dependent)
In a transformation, which is the domain and codomain?
T: R^n -> R^m R^n is the domain R^m is the codomain "T maps R^n to R^m" "T(x) is the image of x under T"
What is the range?
The set of all vectors that are outputs of a mapping T: R^n -> R^m
What are linear transformations?
The multiplication by a matrix
AKA Matrix transformations
“x–>Ax”
“x maps to Ax”
What are the 2 conditions of a linear transformation?
- T(u+v)=T(u)+T(v)
2. T(cu)=cT(u)
What is injective mapping?
A mapping T: R^n -> R^m is one-one if each b in R^m is the image of at most one x in R^n (no doubles on the dots in the right circle)
If T is not injective, T(x)=0 has more than the trivial solution, so its linearly dependent
What is surjective mapping?
A mapping T: R^n -> R^m Is ONTO if each b in R^m is the image of at least one x in R^n (each dot in the right circle is used)
