linear independence

a set of vectors is linearly independent if

• there are no free variables in the solution set to Ax = 0

• A has a pivot in every column

• the vector equation x1v1 + x2v2 + xpvp = 0 has only the trivial solution where x1 = … = xp = 0

any set of vectors containing the 0 vector is

linearly dependent

facts about linear independence

1. pivot columns of a matrix are always linearly independent

2. you cannot have 3 linearly independent vectors in R²

3. a wide matrix cannot have linearly independent columns

True or False: The columns of a matrix with dimensions mxn where m < n must be linearly dependent

True. It is impossible for a matrix with less rows than columns to have a pivot in each column

True or False: set S of vectors contains fewer vectors than there are entries in the vectors, then the set must be linearly independent

False. The 0 vector exists

True or False:Two vectors are linearly dependent if an only if they are collinear

True.

True or False: If S is a set of linearly dependent vectors, then every vector in S can be written as a linear combination of the other vectors in S

False. In order for S to be linearly dependent, only one vector in S needs to be expressible as a linear combination of the others

True or False: The columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution

False. the equation Ax = 0 always admits the trivial solution, whether or not the columns of A are linearly independent.