Linear Algebra Flashcards
Linear Equation
An equation in the form a1x1+a2x2+…+anxn=b where n is a postive integer a1,a2,…,an,b are numbers and x1,x2,…,xn are variables.
LInear System
A list of one or more linear eequations
Solution to a linear system
A solution to one linear system a1x1+a1x2+…+anxn=b is a list of numbers (s1,s2,…,sn) such that a1s1+a2s2+…+ansn is equal to b. A solution to a linear system is a list of numbers that is simultaneously a solution to every equation in the system.
Equivalent Linear Systems
Two linear systems with the same sets of variables and the same set of solutions.
Inconsistent Linear Systems
A linear system with no solution
Matrix
A rectangular array of numbers
Coefficient matrix
For a linear equation with m equations and n variables, the mxn matrix that records the coefficients of the variable.
Augmented matrix
For a linear system with m equations and n variables, the m x (n+1) matrix that records the coefficients of the variables and the constant on the other side of each equation.
Elementary row operator
One of the following operations on a matrix: replace one row with the sum of itself and a multiple of another row, multiply all entries in a row by a fixed number, or swap two rows.
Row equivalent matrices
Matrices that can be transformed into each other by a sequence of row operations.
Leading entry
The first non zero entry in a given row, going left to right.
Echelon form
A matrix is in echelon form it if has these properties: If a row is non-zero, then every row above it is also non-zero, the leading entry in one row is in a column to the right of the leading entry in each row above, if a row is nonzero, then every entry below its leading entry in the same column is zero.
Reduced Echelon Form
A matrix in RREF has 1 as the leading entry in each nonzero row and has no other nonzero entries in the same column as a leading entry in a row.
Pivot Position and Pivot Column
The location containing a leading 1 in the RREF of A
Basic Variable and Free Variable
If A is the augmented matrix of a linear system in x1,x2,…,xn. xi is a basic variable if i is a pivot column of A and i is a free variable if i is not a pivot column of A.
Span of a list of vectors v1,v2,…,vp that spans R^n
Is a set of all linear combinations of the vector.
Linerly independent
The vectors v1,v1,…,vp than spans R^n are linearly independent when x1v1+x2v2+…+xpvp=0 only when x1=x2=…xp=0
If A is a mxn vector
- For each vector b in R^m, Ax=b has a solution.
- A has a pivot in every row.
- Every vector b in R^m is a linear combination of the column in A.
- The span of column A is R^m.
If a matrix have more columns than row
Then, they are linearly dependent
How to determine linear independence
RREF the matrix then if the matrix has a pivot in every column, it is linearly independent.
A single vector v is linearly independent only if
v is not equal to o
A list of vectors in R^n is linearly dependent if
it includes a zero vector, some vector vi is a linear combination of the other vectors and if column>row.
Domain, Codomain and Range
The domain is the input for the function, Codomain is a set that contains the output of the function, and Range is all possible output of the function.
Linear Combination of vectors
Is the vector obtained by adding two or more vectors which are multiplied by scalar values.
Linear Function f=R^n -> R^m
F is a transformation matrix where F(u+v)=F(u)+F(v) and F(cV)=cF(v) for u,v∈ R^n and c∈ R or there is an mxn matrix A such that F has the formula F(v)=av for v∈ R^n
One to One Function f=R^n -> R^m
A function with the property that if f(u) = f(v) for u,v ∈ R^n then u=v which happens if and only if T(v)=Av where A is an mxn matrix and the columns of A are linearly independent meaning A has a pivot in every column.
Onto Function f=R^n -> R^m
A function with the property y ∈ Y then there exist x ∈ X with f(x)=y. T is onto if and only if the span of the column of A is R^m, which happens when A has a pivot position in every row.
Invertible function f=R^n -> R^m
A function such that for every y ∈ Y there is exactly one element x ∈ X such that f(x)=y
Subspace of R^n
is a set H of vectors in Rn such that
1. the zero vector is in H. (0 ∈ H)
2. For every two vectors u and v, the sum u+v is also in H ( u,v ∈ H, u+v ∈ H)
3. If v ∈ H, and c ∈ R, then cV∈ H
Basis of a subspace
is a linearly independent set of vectors where every vector in the subspace can be written as a linear combination of the basis vectors.
Dimension of a subspace
is the number of vectors in any basis for the subspace.
If A is an mxn matrix what is the (a) nullspace of A (b) column space of A (c)rank of A
(a) Nullspace is the set of vectors in R^n such that Av=0
(b) Column space is the space spanned by the column of A it is a subspace of R^m
(c) Rank is the number of pivot columns in A or the dimension of the column space Col A.
Let A be an nxn matrix that is invertible, then,
- There is an nxn matrix A^-1 such that AA^-1=A^-1A=In
- For each b∈ R^n the equation Ax=b has a unique solution
- RREF(A)=In
