Applied Computational Mathematics

1. Matrix Algebra Fundamentals

A matrix A in mathbb{R}^{m imes n} is a rectangular array of numbers arranged in m rows and n columns.

• Matrix Multiplication: For A in mathbb{R}^{m imes k} and B in mathbb{R}^{k imes n}, the product C = AB in mathbb{R}^{m imes n} has entries:

• Transpose Properties:

• Symmetric & Skew-Symmetric:
• Symmetric: A^T = A
• Skew-Symmetric: A^T = -A (all diagonal entries a_ii = 0)
• Any square matrix A can be uniquely expressed as:

(Sum of a symmetric matrix and a skew-symmetric matrix).

2. Orthogonal & Unitary Matrices

A real square matrix Q in mathbb{R}^{n imes n} is orthogonal if:

• det(Q) = pm 1.
• Preserves vector lengths and inner products: |Q mathbf{x}| = |mathbf{x}|.
• In complex matrices, U^* U = I defines a unitary matrix (U^-1 = U^*).

3. Solved University Numerical

Problem: Express A = egin{bmatrix} 3 & 5  1 & -1 end{bmatrix} as the sum of a symmetric and skew-symmetric matrix.

1. Symmetric part P = rac{1}{2}(A + A^T):

1. Skew-symmetric part Q = rac{1}{2}(A - A^T):

1. Verification: P + Q = egin{bmatrix} 3 & 5  1 & -1 end{bmatrix} = A.

1. Elementary Row Operations (ERO)

On any matrix A in mathbb{R}^{m imes n}, three elementary operations preserve row equivalence (A sim B):

1. R_i leftrightarrow R_j: Interchange of any two rows.
2. R_i o k R_i ($k

eq 0): Multiplication of rowiby a non-zero scalark$.

1. R_i o R_i + k R_j: Adding k times row j to row i.

2. Row Echelon Form (REF) Criteria

A matrix is in Row Echelon Form if:

• All zero rows (if any) are strictly at the bottom.
• The leading entry (pivot) of each non-zero row is strictly to the right of the pivot in the row above it.
• All entries in a column below a leading pivot are zero.

Rank of Matrix ho(A):

3. Step-by-Step Gauss Elimination Example

Solve the system:

Augmented matrix [A | B]:

By back substitution:

1. Vector Space & Subspace Axioms

Let V be a vector space over field mathbb{R}. A non-empty subset W subseteq V is a subspace iff:

2. Linear Independence & Basis

A set {mathbf{v}₁, mathbf{v}₂, dots, mathbf{v}_k} is linearly independent if:

• A basis of V is a linearly independent spanning set.
• The dimension dim(V) is the number of vectors in any basis.

3. Rank-Nullity Theorem (High-Yield Exam Proof)

For any linear transformation T: mathbb{R}ⁿ o mathbb{R}^m represented by an m imes n matrix A:

where ext{rank}(A) = dim( ext{Col}(A)) and ext{nullity}(A) = dim( ext{Null}(A)).