Exam 1 Flashcards
Matrices and Vectors
Linear Equation
an equation that can be written as:
ax1+bx2+…..nxn
where a,b..n are real or complex numbers
Linear System
One of more linear equations involving the same variables.
Must have
1. No solution (Consistent)
2. 1 solution (Consistent)
3. Infinite Solutions (Inconsistent)
matrix (elementary row operations)
- Interchange
- Scaling
- Replacement
Row Equivalence
Two matrices are row equivalent if there is a series of elementary row operations that transforms one matrix into another
This means that the matrices associated linear system has the same solution set
Echelon Form
- Non Zero Rows are above rows of all 0s
- Traingle Shape – Leading rows go down left to right
- Below leading entry are zeroes
RRef : Row Reduced Echelon Form
What:
1. Leading Entry is 1
2. Leading entry is only non zero entry
Note: RReF is UNIQUE
Use:
1. Determining qualities of variables, determining solution sets
Pivot Position
What: Corresponds to the position of a leading 1. A pivot column contains a pivot solution.
Use:
Basic Variables (have a solution essentially) correspond to pivot columns
Free variables do not correspond to pivot columns and can be any value
A linear system is consistent if…
it’s corresponding matrix has no row of the form 000 b where b does not equal 0
Consistent system has a solution (1 or infinite)
Why relevant: The existence of a free variable does NOT mean infinite solutions + best way to determine no solution
Vector
- A matrix with one column
- Two vectors are equal if and only if thier entries are the same
- zero vector has zeroes in all positions
Vector Operations
- Scalar Multiplication
- Vector Addition
- Parellelogram Rule
Linear Combo
Combination of a set of vectors with a set of scalars as weights
relevant: for matrices..
span
what: the set of all linear combinations of v1, v2,v3
why: where can we go using these vectors?
matrix equations
A x> = b> only has a solution if b is a linear combination of the columns of A (think of how multiplication works)
Ax = 0
-homogenous if it can be written in that form
- always at least one solution (trivial solution) x = 0
- non trivial solution (one free variable)
The consistent equation with Ax = b, let p be a solution
the solution set is w = p + vh where vh is any solution of the homogenous equation
repercussions: if there is only a trivial solution, then Ax = b is the unique solution
Linear Independence
A set of vectors is linearly indepedent if none of the vectors in the set are linear combinations of the others
- Linearly independent vectors only have the trivial solution
- Does not contain the zero vectors
- More vectors than entries in vectors (more columns than rows)
Given an mxn matrix, for each b in Rm Ax = b has a solution
- b must be a linear combo of A
- columns of A span Rm
- pivot position in each row
Codomain
all possible outputs
range
all outputs
Onto
Range and Codomain are the same
rn –> rm
Every b in rm is the image of at least one x in rn
one to one
every b in Rm is the image of at MOST on x in Rn
Linear Transformation requirement
- vector addition
- scalar multiplication
Linear Transformation Properties
- 1-1 if and only if T(x) = 0 has only the trivial solution (this means that A is linearly independent)
- T maps Rn onto Rm if and only if the columns of A span Rm
Invertibility
AC = I = CA
- A and its inverse are invertible
- Must be a square matrix, no free variables
- (AB)-1 = B-1*A-1
- Must be one to one and onto
Vector Space
A non empty set V of objects which are subject to 10 axioms (on cheat sheet)
1. closed under addition
2. addition is commutative
3. addition is associative
4. closed under multiplication
5. has zero vector
6. an additive inverse exists
7. scalars distrivute
8. vecors distribute
9. associativity w scalars
10, scalar identity
Subspace
- zero vector in H
- closed under addition
- closed under multiplication
Null Space
the solution set of A where x = 0
This is a subspace
Column Space of A
The span of the column of A
The pivot columns of a matrix A are the basis of Col (A)
basis
a set of linearly independent vectors that span H
