Midterm 1 Linear Algebra

Requirements for a basis

1) linear independent

2) span entire vector space

Linear Combination

multiplying a vector by a constant (scaling) or adding vectors

Definition of a span

collection of all vectors made from linear combinations (scaling/adding) of the original vector)

requirements of a vector subset

1) includes the 0 vector

2) closed under vector addition

3) closed under scalar multiplication

Requirements to be a linear map

1) includes 0 vector

2) closed under vector addition

3) closed under scalar multiplication

What is mapping a vector ?

applying a function or transformation to produce a new output vector

Definition of a kernel (null space)

all input vectors that get mapped to the zero vector in the output space

definition of One-to-one and onto

one to one means only one output for every input (has pivot columns) and onto is if the range fills the entire codomain (pivot rows)

Requirements for linear independence

when in RREF, if every column had a pivot and there are no free variables

Basis for a kernel

set up a matrix with the last row being all zeros and row reduce