Linear algebra final

What does it mean for a collection {v1, …,vp} of vectors in R^n to be linearly independent?

If x1*v1+….+xp*vp=0 has only the trivial solution, x1=….=xp=0

If v1,….,vp are vectors in R^n, give the precise definition of Span{v1,…,vp}

Span {v1,…vp} is the collection of all linear combinations of v1,…,vp

What does it mean for a transformation to be one-to-one?

If Tx=b has at most one solution x in R^n

What does it mean for a transformation to be onto?

If Tx=b has at lease one solution x in R^n

Domain

Number of columns

Codomain

Number of rows

General Solution

Parametric Form

One to one

• At most one solution

• Columns

• Has the same outputs as inputs or more outputs

Onto

• At least one solution

• Rows

• Has the same outputs as inputs or fewer outputs

When do columns span n?

When there is a pivot in every row

Columns are linearly independent if

• Only the trivial solution

• Pivot in every column

• No free variables

Invertible?

Has no free variables