1.7 Linear Independence

linear independence

an indexed set of vectors [v1, … vp] in allRⁿ is said to be independent if the vector equation

x1v1 + x2v2 +…xpvp = 0

has only the trivial solution

linear dependence

an indexed set of vectors [v1,…vp] in allRⁿ is said to be linearly dependent if there exists wights c1,…cp not all zero, such that

c1v1+c2v2+…cpvp = 0

to determine if set of vectors is linearly independent, you must determine if x1v1+x2v2+… = 0 has a ? solution

nontrivial

if there is a free variable in the matrix, then the vectors are linearly ?

dependent

this is because it means there is a nontrivial solution to the homogeneous system Ax = 0,

to find dependence relation among v1, v2, and v3, we need to find x1,x2,x3 not all zero that satisfy the equation:

x1v1+x2v2+x3v3=0

THEN, choose a value for the free variable and find the other nonzero x1, x2, xn values and create a dependence relation

the columns of martrix A are linearly ? if and only if the equation Ax=0 has only the trivial solution

independent

if matrix has no free variables (pivot for each row), then Ax=0 has only the ? solution (x=0)

trivial

a set of one vector [v] is linearly independent if and onlyif v is not the ? vector

zero

set of vectors is linearly dependent if an only if x1v1 + x2v2 + x3v3 = 0 has a ? solution / needs a ? variable

nontrivial, free

a set of two vectors is linearly dependent if at least one of the vectors is a ? of the other

multiple

an indexed set of two or more vectors is linearly dependent if an only if at least one of the vectors in S is a ___ ___ of the others

linear combination

if a set contains more vectors than there are entries in each vector then the set is __ __

linearly dependent

if a set of vectors in allRⁿ contains the zero vector, then the set is ___ ___

linearly dependent

if there are nonzero solutions of Ax=0, it is linearly ?

dependent

The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution.

FALSE

Every homogeneous matrix equation has the trivial solution, but not every matrix has linearly independent columns.

(T/F) If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.

FALSE

at least one vector in a linearly dependent set must be a linear combination of the others, but it need not be the case that every vector in the set is a linear combination of the others.

If x and y are linearly independent, and if z is in   Span  {x, y}, then {x, y, z} is linearly dependent.

TRUE