Math 313 Review

Question 1

Homogeneous system

Answer 1

System is homogenous if can be written Ax=0

Question 2

Trivial solution

Answer 2

Solution where x=0

Question 3

Nontrivial solution

Answer 3

For Ax=0 if the equation has at least one free variable

Question 4

Matrix Equation Theorem

Answer 4

1. For each b in R^m, the equation Ax=b has a solution
2. Each b in R^m is a linear combination of the columns of A.
3. The columns of A span R^m
4. A has a pivot position in every row

Question 5

Span of vectors

Answer 5

If augmented matrix has a zero row, it is within the span. If it becomes consistent, then it is linear independent and not in the span

Question 6

Existence and uniqueness theorem

Answer 6

LInear system is consistent if we don’t get a row where 0=x.

Question 7

Consistent systems

Answer 7

Have a unique solution with no free variables

Have infinite solution if it has one free variable

Question 8

Pivot position

Answer 8

Entry in matrix that has leading 1 in RREF

Question 9

Pivot column

Answer 9

Column of A that has a pivot position

Question 10

Echelon form

Answer 10

1. All nonzero rows are above any rows with zeros
2. Each leading entry of row is in a column to the right of leading entry row above it
3. All entries in a column below leading entry are zeros

Question 11

Reduced row echelon form

Answer 11

1. Echelon Form
2. Leading entry in each nonzero row is 1
3. Each leading 1 is the only nonzero entry in its column

Question 12

Consistent

Answer 12

System with at least one solution

Question 13

Inconsistent

Answer 13

System with no solutions

Question 14

Linear dependence

Answer 14

If a set contains more vectors than there are entries, the set is linearly dependent. For example, 4 vectors with only 3 entries.

Question 15

T: R^n -> R^m

Answer 15

One-to-one, m >=n

Onto R^m, m<= n

Question 16

If vector has only trivial solution

Answer 16

The vectors are linearly independent