Linear Algebra

a system of linear equations has either one solution or infinitely many solutions

no solution

1. augment the matrix
2. reduce to triangular form using row operations

a location in matrix A that corresponds to a leading 1 in the reduced echelon form of A

use the free variables as the parameters for describing a solution set

the set of all vectors with two entries

vectors in R3 are 3×1 column matrices with three entries

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

1. augment the matrix to [a1 a2 b] 2. if solution is consistent, 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

1. determine whether the vector equation x1v1 + x2v2 + … + xpvp = b
2. equivalently, determine whether the augmented matrix [v1 … vp b] has a solution

Ax is defined only if the number of columns of A equals the number of entries in x

Ax = b

x1a1 +x2a2 + … + xnan = b

exists if and only if b is a linear combination of the columns of A

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

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

the zero solution (for Ax = 0, x = 0)

a nonzero vector x that satisfies Ax = 0

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

x = su + tv (s,t in R)

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

1. perform row operations on [A b] 2. express the solution in terms of the free variables

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

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)

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

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)

the columns of matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution (no free variables)

1. augment the matrix to [A 0] 2. row reduce
3. if there are no free variables, the columns of 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

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

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)

a square n x n matrix whose non diagonal entries are zero

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

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

1. write down theorem 4 from pg 103
2. if ad-bc = 0, then A is not invertible
3. det A = ad-bc
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

1. find the inverse of A
2. multiply by the b (the right hand side of the equals sign)

write down theorem 6 from pg 105

1. augment the matrix to [A I] 2. row reduce to reduced echelon form

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

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

if v1 and v2 are in Rn and H = span{v1, v2}, then H is a subspace of Rn

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}

1. augment the matrix to [A b] 2. row reduce
3. if solution is consistent, b is in Col 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

a basis for a subspace H of Rn is a linearly independent set in H that spans H

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

1. look this up

definition pg 154

dim H is the number of vectors in any basis for H

the rank is the dimension of the column space of A

dim Nul A = the number of free variables in Ax = 0

if a matrix A has n columns, then rank A + dim Nul A = n

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

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

det A is the product of the entries on the main diagonal of A

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

a square matrix A is invertible if and only if det A does not equal 0

if A is a square matrix, then det A^T = det A

if A and B are square matrices, then det AB = (det A)(det B)

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

let A be an invertible square matrix
A^-1 = 1/(det A) * adj A

see page 190

see page 193

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

1. write W as a set of linear combinations
2. use the vectors in the spanning set as the columns of A

pick any column of A that is nonzero

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

the set of all u in V such that T(u) = 0