MATH240 Linear Algebra Exam 2

Raluca Rosca's MATH240 Exam 2 preparation (necessary formulas and rules)

$Rank(B)$ bounds

$$Rank(B) \leq min(n, m)$$

Area

$$|det[u v]|$$

Dimension formula for subspaces

$$dim(H+K) = dim(H) + dim(K) - dim(H \cap K)$$

Rank-Nullity Theorem

Rank (Pivot cols) + Nullity (Free var cols) = Total cols

Basis of $Col A$

1. Row reduce

2. Find pivot columns

3. Take from original matrix $A$

Basis of $Row A$

1. Row reduce

2. Take non-zero rows

Basis of $Nul A$

1. Row reduce

2. Identify free variables

3. Write solution in parametric-vector form (free x vector)

Find change of basis matrix given vectors in both spaces

1. Set up matrix with target matrix on left and translatable matrix on right

2. Find $RREF$ of left

3. Right matrix is change of basis

Dimension of a span

Number of linearly independent vectors in a set

Kernel

Null Space of a linear transformation

What happens when a square matrix has no zero eigenvalues?

It is invertible, and vice versa

Eigenvector formula

$$det(A - \lambda I)$$

How to check if a matrix is an isomorphism?

Square matrix with a non-zero determinant (invertible).