notes / updates | linear algebra

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Dec 6, 2025 | A list of topics

Chapter 1
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Find ref and rref
Solving vector equations using the augmented matrix and row reduction.
Expressing solutions of a vector equation.
Linear (in)dependence.
Working with Linear Transformations (geometrically)
Working with Linear Transformations (the matrix)
Skip: 1.10

Chapter 2
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Matrix operations
The inverse
The invertible matrix theorem
LU factorization.
Subspaces of R^n
Dimension and Rank
Skip: 2.4, 2.6, 2.7

Chapter 3
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The Determinant and its properties
Cramer's Rule, Volume, Linear Transformations
kernel (nullspace); colspace (image space)
dimensions of kernel, colspace, and rank of a matrix

Chapter 5
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Eigenvectors, eigenvalues, the characteristic equation.

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October 24, 2025 | LU decomposition with SageMath

   
  #Example 2 | Section 2.5 
A = matrix([[2,4,-1,5,-2],[-4,-5,3,-8,1],[2,-5,-4,1,8],[-6,0,7,-3,1]])

P, L, U = A.LU(pivot='nonzero')
print(U)
print(L)

#check
print(A == L*U)


# Practice Problem | Section 2.5

A = matrix([[2,-4,-2,3],[6,-9,-5,8],[2,-7,-3,9],[4,-2,-2,-1],[-6,3,3,4]])

P, L, U = A.LU(pivot='nonzero')
print(U)
print(L)

print(A == L*U)
   
    

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