Chapter 2

Question 1

Def: consistent/inconsistent

Answer 1

If a system has at least one solution, it is said to be consistent
Otherwise, it is said to be inconsistent

Question 2

Def: equivalent

Answer 2

Two systems of equations with the same solution set are said to be equivalent

Question 3

Def: coefficient and augmented matrix

Answer 3

The augmented matrix is the one that has the | solutions part
Coeff: [A]
Augment: [A|b]

Question 4

Def: elementary row operations

Answer 4

1) multiply row by non scalar
2) add a multiple of one row or another
3) swapping two rows

Question 5

Def: row equivalent

Answer 5

Two matrices are said to be row equivalent if there exists a sequence of elementary row operations that transform A into B
A~B

Question 6

Def: reduced row echelon form

Answer 6

RREF

1) zero rows on the bottom
2) first nonzero entry in each row is 1
3) leading one in each non zero row is to right of previous
4) leading one is only nonzero entry in column

Question 7

Theorem 2.2.2

Answer 7

If A is a matrix, then A has a unique RREF R

Question 8

Gauss-Jordan Elimination

Answer 8

Basically put matrix into RREF

Question 9

Def: free variable

Answer 9

Let R be the RREF of a coefficient matrix of a system of linear equations
If the column of R does not contain a leading one, then we call xj a free variable

Question 10

Def: homogenous system

Answer 10

A system of linear equations is said to be a homogeneous system if the right-hand side only contains zeros
[A|0]

Question 11

Theorem 2.2.3

Answer 11

The solution set of a homogeneous system of m linear equations in n variables is a subspace of R^n

Question 12

Def: solution space

Answer 12

The solution set of a homogeneous system is called the solution space of the system

Question 13

Def: rank

Answer 13

The rank of a matrix is the # of leading ones
Demoted rank(A)

Question 14

Theorem 2.2.4

Answer 14

For any mxn matrix A we have rank(A)

Question 15

Theorem 2.2.5 (system rank theorem)

Answer 15

Let A be the coefficient matrix of a system of m linear equations in n unknowns [A|b]

1) rank(A) < rank([A|b]) iff the system has no solution
2) if [A|b] has a solution, then the system has n-rank(A) free variables
3) rank(A)=m iff [A|b] has a solution for every bER^m

Question 16

Theorem 2.2.6 (solution theorem)

Answer 16

Let [A|b] be the augmented matrix, [C|d] be the matrix in RREF
If rank(A) = rank(C) = k LESS THAN n, the so,union set of the augmented matrix is either an empty set or an n-k dim plane in Rn

Question 17

Theorem 2.2.7 (theorem from class)

Answer 17

1) {v1,…,vk}CR^m is linearly independent iff the rank of the mxk matrix [v1…vk] is k
2) {v1,…,vm}CR^m is linearly independent iff Span{v1,…,vm}=R^m

Question 18

Another theorem~

Answer 18

If A is an mxn matrix, then rank(A) less or equal m AND less or equal to n