Jan 24

• Introduce nullspace and columns space
• Draw picture of domain, range, kernel, codomain

Theorem related to linear independence

Let S = {a₁, …, a_n} ⊆ ℝ^m be a set of vectors. Let A be the m × n matrix formed by writing the elements of S as columns. Let T : ℝⁿ → ℝ^m be the linear transformation defined by T(x) = Ax.

• S is linearly independent
• Ax = 0 has only the trivial solution
• For any b, Ax = b has either no solution or exactly one solution.
• null(A) = {0}
• T is one-to-one
• ker(T) = {0}

Theorem related to spanning

Let S = {a₁, …, a_n} ⊆ ℝ^m be a set of vectors. Let A be the m × n matrix formed by writing the elements of S as columns. Let T : ℝⁿ → ℝ^m be the linear transformation defined by T(x) = Ax.

• S is spanning
• Ax = b has a solution for any b
• col(A) = ℝ^m
• T is onto
• range(T) = ℝ^m

Theorem related to the square case

Let S = {a₁, …, a_n} ⊆ ℝⁿ be a set of vectors. Let A be the n × n matrix formed by writing the elements of S as columns. Let T : ℝⁿ → ℝⁿ be the linear transformation defined by T(x) = Ax.

• S is a basis
• S is linearly independent
• S is spanning
• Ax = b always has a unique solution
• col(A) = ℝⁿ
• null(A) = ℝⁿ
• T is invertible
• A is invertible

3.2 Matrix Algebra

Matrix multiplication is weird

• AB ≠ BA
• Explain what AB = 0 means in terms of columnspace and nullspace

Tranpose of a matrix

• Teach how to tranpose
• (A + B)^t = A^t + B^t
• (sA)^t
• (AC)^t = C^tA^t

Diagonal matrices and upper triangular matrices is a thing

• Give definition
• The product of digaonl is diagonal. Discuss the effects of multiplying a matrice by a diagonal matrix
• The product of upper triangulars is upper triangular

Powers of matrices is a thing

• Powers of diagonal is easy
• Wouldn’t it be great if A = UDU^− 1

3.3 Inverses

• Explain what an inverse is.
• Derive inverse formula.