# Linear Algebra

a system of linear equations has either one solution or infinitely many solutions
consistent
no solution
inconsistent
We will write a custom essay sample on
Linear Algebra
or any similar topic only for you
1. augment the matrix
2. reduce to triangular form using row operations
determine if a system of linear equations is consistent or inconsistent
a location in matrix A that corresponds to a leading 1 in the reduced echelon form of A
pivot position
use the free variables as the parameters for describing a solution set
parametric description of a solution set
the set of all vectors with two entries
R2
vectors in R3 are 3×1 column matrices with three entries
R3
given vectors v1, v2, …, vp in Rn and scalars c1, c2, …, cp, the vector y is called a linear combination and is defined by y = c1v1 + … + cpvp
linear combination (y)
1. augment the matrix to [a1 a2 b] 2. if solution is consistent, b is a linear combination of a1 and a2
determine wether b is a linear combination of a1 and a2
the span of {v1,…,vp} is a set of all linear combinations of v1…vp
must contain the zero vector
span
1. determine whether the vector equation x1v1 + x2v2 + … + xpvp = b
2. equivalently, determine whether the augmented matrix [v1 … vp b] has a solution
determine if b is in span{v1,…,vp}
Ax is defined only if the number of columns of A equals the number of entries in x
matrix multiplication
Ax = b
matrix equation
x1a1 +x2a2 + … + xnan = b
vector equation
exists if and only if b is a linear combination of the columns of A
solution of Ax = b
let A be an mxn matrix
1. for each b in Rm, Ax = b has a solution
2. each b in Rm is a linear combination of the columns of A
3. the columns of A span Rm
4. A has a pivot position in every row
coefficient matrix theorem
1. can be written in the form Ax = 0 where A is an mxn matrix and 0 is the zero vector in Rm
2. always has at least one solution
3. Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable
homogenous linear system
the zero solution (for Ax = 0, x = 0)
trivial solution
a nonzero vector x that satisfies Ax = 0
nontrivial solution
1. let A be the matrix of coefficients of the system
2. row reduce the augmented matrix [A 0] to echelon form
3. determine if a free variable exists
4. to describe the solution set, continue row reduction to reduced echelon form
determine if the homogenous system has a nontrivial solution
x = su + tv (s,t in R)
parametric vector equation
the general solution (many solutions) can be written in parametric vector form as one vector plus an arbitrary linear combination of vectors that satisfy the corresponding homogenous system
solutions of nonhomogenous systems
1. perform row operations on [A b] 2. express the solution in terms of the free variables
describe all solutions of Ax = b
1. a set of vectors {v1,…,vp} in Rn is said to be linearly undefended if the vector equation x1v1 + x2v2 + … + xpvp = 0 has only the trivial solution (x=0)
2. pivot in every column
3. one-to-one
linearly independent
1. the set of vectors {v1,…,vp} is linearly dependent if there exist weights c1,…,cp (not all zero) such that c1v1 + c2v2 + … + cpp = 0
2. a nontrivial solution exists (at least one free variable)
linearly dependent
1. A = [v1 v2 v3] 2. augment the matrix to [A 0] and row reduce
3. if there is at least one free variable, the set is NOT linearly independent
determine if the set {v1, v2, v3} is linearly independent
1. completely row reduce the augmented matrix and write the new system of linear equations
2. choose any nonzero value for the free variables
3. substitute into equation 1 (pg 56)
find a linear dependence relation among v1, v2, and v3
the columns of matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution (no free variables)
linear independence of matrix columns
1. augment the matrix to [A 0] 2. row reduce
3. if there are no free variables, the columns of A are linearly independent
determine if the columns of matrix A are linearly independent
1. check whether at least one of the vectors is a scalar times the other (only applies to sets of two vectors)
2. if so, the vectors are linearly dependent
3. if not, the vectors are linearly independent
determine if the sets of vectors are linearly independent
1. a set of two or more vectors is linearly dependent if and only if at least of of the vectors is a linear combination of the others
2. if a set contains more vectors than there are entries in each vector, then the set is linearly dependent
3. if a set contains the zero vector, then the set is linearly dependent
linear dependency of sets of two or more vectors
1. check to see if the number of vectors is greater than the number of entries in each vector
2. check to see if the set has the zero vector
3. compare the corresponding entries of the two vectors (determine if one is a multiple of the other)
determine if the set of vectors is linearly dependent
a square n x n matrix whose non diagonal entries are zero
diagonal matrix
the sum of A + B is defined only when A and B are the same size
1. the number of columns of A must match the number of rows in B in order for a linear combination such as Ab1 to be defined
2. AB has the number of rows as A and the number of columns of B
3. AB is the matrix A times each column of B –> AB = [Ab1 Ab2 … Abp] 4. does not matter how we group the matrices when computing the product, as long as the left-to-right order of the matrices is preserved
5. AB usually does not equal BA
matrix multiplication
1. given an m x n matrix, the transpose of A (A^T) is the n x m matrix whose columns are formed from the corresponding rows of A
2. (A^T)^T = A
3. (A+B)^T = A^T + B^T
4. (AB)^T = B^T*A^T
the transpose of a matrix
1. write down theorem 4 from pg 103
2. if ad-bc = 0, then A is not invertible
4. if A is an invertible square matrix, then for each b in Rn, the equation Ax = b has the unique solution x = (A^-1)*b
the inverse of a matrix
1. find the inverse of A
2. multiply by the b (the right hand side of the equals sign)
use the inverse of the matrix A to solve the system of linear equations
write down theorem 6 from pg 105
properties of an invertible matrix
1. augment the matrix to [A I] 2. row reduce to reduced echelon form
algorithm for finding A^-1
if…
1. A is a square matrix
2. A can be reduced to an I matrix
3. the number of pivot positions = number of rows or number of columns, A is invertible
4. Ax = 0 only has the trivial solution, A is invertible
5. the columns of A are linearly independent, A is invertible
6. the equation Ax = b has exactly one solution
7. the columns of A span Rn
8. A^T is an invertible matrix
then A is invertible
use the Invertible Matrix Theorem to decide if A is invertible
a subspace of Rn is any set H in Rn that has three properties:
1. the zero vector is in H
2. for each u and v in H, the sum u+v is in H
3. for each u in H and each scalar c, the vector cu is in H
subspace
if v1 and v2 are in Rn and H = span{v1, v2}, then H is a subspace of Rn
subspace and span
1. the column space of a matrix A is the set Col A of all linear combinations of the columns of A
2. if A = [a1 … an] then Col A is the same as Span{a1…an}
column space
1. augment the matrix to [A b] 2. row reduce
3. if solution is consistent, b is in Col A
determine whether b is in the column space of A
1. the null space of a matrix A is the set Nul A of all solutions of the homogenous equation Ax = 0
2. when A has n columns, the solutions of Ax = 0 belong to Rn and the null space of A is a subset ofRn
3. the null space of an m x n matrix is a subspace of Rn
null space
a basis for a subspace H of Rn is a linearly independent set in H that spans H
basis for a subspace
1. augment the matrix to [A 0] 2. row reduce
3. write the solution of Ax = 0 in parametric vector form
4. express the solution in terms of the free variables
5. u, v, w are a basis for Nul A
find a basis for the null space of the matrix A
1. look this up
find a basis for the column space of the matrix A
definition pg 154
coordinate systems
dim H is the number of vectors in any basis for H
dimension of a subspace
the rank is the dimension of the column space of A
rank
dim Nul A = the number of free variables in Ax = 0
determine the dimension of Nul A
if a matrix A has n columns, then rank A + dim Nul A = n
rank theorem
let H be a p-dimensional subspace of Rn. any linearly independent set of exactly p elements in H is automatically a basis for H. any set of p elements of H that spans H is automatically a basis for H
the basis theorem
if A is an invertible n x n matrix,
1. the columns of A form a basis of Rn
2. Col A = Rn
3. dim Col A = n
4. rank A = n
5. Nul A = {0}
6. dim Nul A = 0
the invertible matrix theorem (cont.)
det A is the product of the entries on the main diagonal of A
determinant of a triangular matrix
let A be a square matrix
1. if a multiple of one row of A is added to another row to produce a matrix B, then det B = det A
2. if two rows of A are interchanged to produce B, then det B = -det A
3. if one row of A is multiplied by k to produce B, then det B = k * det A
properties of determinants
a square matrix A is invertible if and only if det A does not equal 0
invertibility and determinants
if A is a square matrix, then det A^T = det A
the determinant of a transpose
if A and B are square matrices, then det AB = (det A)(det B)
multiplicative property of determinants
let A be an invertible square matrix. for any B in Rn, the unique solution x of Ax = b has entries given by
page 177
Cramer’s rule
let A be an invertible square matrix
A^-1 = 1/(det A) * adj A
see page 190
vector space
see page 193
subspace
1. find a general solution to Ax = 0 in terms of the free variables
2. row reduce the augmented matrix [A 0] to reduced echelon form
3. {u v w} is a spanning set for the null space of A
find a spanning set for the null space of the matrix A
1. write W as a set of linear combinations
2. use the vectors in the spanning set as the columns of A
find a matrix A such that W = Col A
pick any column of A that is nonzero
find a nonzero vector in Col A
1. row reduce [A 0] 2. solve for the variables in terms of the free variables
3. assign a nonzero value to the free variables
find a nonzero vector in Nul A
the set of all u in V such that T(u) = 0
kernel

#### New Essays

×

Hi there, would you like to get such a paper? How about receiving a customized one? Check it out