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Linear Algebra

  1. Vectors
    1. Scalars
    2. Vectors vs Sets
    3. Addition and Subtraction
    4. Scalar Multiplication
    5. Zero Vectors
    6. Linear Combinations
    7. Real Dot Product
    8. Length of a Vector
    9. Orthogonal Vectors
    10. Parallel Vectors
    11. Angle Between Vectors
    12. Unit Vectors
  2. Matrices
    1. Notation
    2. Indexing
    3. Submatrices
    4. Matrix-by-Vector Product
    5. Addition and Subtraction
    6. Scalar Multiplication
    7. Transpose
    8. Symmetries
    9. Matrix Multiplication
    10. Identity Matrix
    11. Non-Negative Integer Powers
    12. Reverse Order Law of Transposition
  3. Linear Systems
    1. Inverse Matrices
    2. Singular Matrices
    3. Linear Dependence
    4. Solutions
  4. Planes
    1. Vector Cross Product
  5. Gaussian Elimination
Linear Algebra β€ΊMatrices β€ΊSubmatrices

Submatrices

A submatrix is always defined with respect to another matrix. You always create a β€œsubmatrix of another matrix”.

You obtain a submatrix of some matrix AAA by removing an arbitrary combination of rows and columns from said matrix AAA.

For instance, take

A=[86βˆ’12βˆ’58βˆ’357βˆ’2βˆ’83].A = \begin{bmatrix} 8 & 6 & -1 \\ 2 & -5 & 8 \\ -3 & 5 & 7 \\ -2 & -8 & 3 \end{bmatrix}.A=​82βˆ’3βˆ’2​6βˆ’55βˆ’8β€‹βˆ’1873​​.

We obtain the matrix

B=[2βˆ’5βˆ’2βˆ’8],B = \begin{bmatrix} 2 & -5 \\ -2 & -8 \end{bmatrix},B=[2βˆ’2β€‹βˆ’5βˆ’8​],

which is a submatrix of AAA, by removing the third column and the first and third rows from AAA.

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Indexing of a Matrix
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Multiplying a Vector by a Matrix
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