Math Test #2

REF requrements

1. all zero rows at bottom

2. all leading entries are left of leading entries below them

RREF requirements

1. zero rows at bottom

2. all leading entries are left of leading entries below them

3. all leading entries are 1

4. all other values in columns with leading entries are 0

rank

number of leading entries (or number of nonzero rows, excluding the constants) in an REF or RREF matrix

consistent

having at least one solution

homogeneous

system whose constants are all 0; has at least one trivial solution

span

the set of all vectors that can be reached via a linear combination of a particular set of vectors

linearly dependent

set of vectors that can make a linear combination of 0

linearly independent

set of vectors that is not linearly dependent

theorem for finding if a system has a solution

a system with the augmented matrix [A|b] is consistent iff b is a linear combination of the columns of A

five ways to determine linear dependence/independence

1. VECTORS/COMPONENTS: dependent if # of vectors > # of components

2. INSPECT: dependent if there are scalars so that c₁v₁ + c₂v₂ + … c_nv_n = 0

3. A LIN. COMBO: dependent iff one can be expressed as a linear combination of the others

4. COLUMNS: dependent iff vectors are columns of matrix A such that [A|0] has a non-trivial solution

5. ROWS: dependent iff m vectors are rows of matrix A such that rank_A < m

determining if a point is within span

1. set vectors as columns of A and point as b

2. if [A|b] is consistent, b is within span of vectors

   • We’re essentially solving for the coefficients by which we would multiply the vectors to get b. If there are nonzero possibilities for those coefficients, b is within the span of the vectors.

determining currents in a circuit

1. develop system of equations using:

   • conservation of currents (I)

   • voltage drops: V = IR and ∑voltage drops = total voltage

2. solve, using parameters if necessary

determining the flow of edges in a network

1. develop system using conservation of flow at nodes

2. solve; use parameters if necessary