Precise Linear Algebra Pocket Guide | Generated by AI
As your engineering-focused tutor, this guide emphasizes verifiable, index-explicit computations for matrices \( A = [a_{ij}]_{i=1}^m, j=1^n \in \mathbb{R}^{m \times n} \) (rows indexed by \( i \), columns by \( j \), 1-based). All steps are shown with indices for clarity; examples annotate entries inline. Review via row reduction and eigenvalue solves on 2×2/3×3 cases.
1. Matrix Operations
- Addition: \( (A + B){ij} = a{ij} + b_{ij} \) for all \( i,j \).
- Scalar Multiplication: \( (cA){ij} = c a{ij} \) for scalar \( c \), all \( i,j \).
- Matrix Multiplication (if \( m \times p \) and \( p \times n \)): \( (AB){ij} = \sum{k=1}^p a_{ik} b_{kj} \) for all \( i=1^m \), \( j=1^n \).
- Transpose: \( (A^T){ij} = a{ji} \); thus \( (AB)^T_{ij} = \sum_k b_{ki} a_{kj} = (B^T A^T)_{ij} \).
- Inverse (for square \( n \times n \), \( \det A \neq 0 \)): \( A^{-1} \) satisfies \( \sum_k a_{ik} (A^{-1}){kj} = \delta{ij} \) (Kronecker delta: 1 if \( i=j \), 0 else). Properties: \( (AB)^{-1}{ij} = \sum_k (B^{-1}){ik} (A^{-1}){kj} = (B^{-1} A^{-1}){ij} \); \( (A^T)^{-1}{ij} = \sum_k (A^{-1}){ki} (A^T){kj} ? Wait, no: (A^{-1})^T{ij} = (A^{-1}){ji} \), so \( [(A^T)^{-1}]{ij} = (A^{-1})_{ji} \).
Example (2×2 Inverse Annotation): Let \( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \). Then \( A^{-1} = \frac{1}{\det A} \begin{pmatrix} a_{22} & -a_{12} \ -a_{21} & a_{11} \end{pmatrix} \), where \( \det A = a_{11} a_{22} - a_{12} a_{21} \).
2. Determinants
- Definition: For square \( A \), \( \det A \) via cofactor expansion along row \( i \): \( \det A = \sum_{j=1}^n a_{ij} C_{ij} \), where minor \( M_{ij} \) is submatrix deleting row \( i \) and column \( j \) (so \( M_{ij} = [m_{pq}] \) with \( p=1^{n-1} \setminus i \), \( q=1^{n-1} \setminus j \), relabeled 1-based), cofactor \( C_{ij} = (-1)^{i+j} \det(M_{ij}) \).
- Properties:
- \( \det(AB) = \det A \cdot \det B \); \( \det(A^T) = \det A \) (since expansion symmetric).
- \( \det(cA) = c^n \det A \).
- Row swap: \( \det \) multiplies by -1; add multiple of row \( k \) to row \( i \neq k \): unchanged; scale row \( i \) by \( c \): multiplies by \( c \).
- \( \det I = 1 \) (diagonal 1s); singular if \( \det A = 0 \) (rank < n).
- Adjugate: \( \adj(A){ij} = C{ji} = [C^T]{ij} \), where \( C = [C{pq}] \). Inverse: \( A^{-1} = \frac{1}{\det A} \adj A \), so \( (A^{-1}){ij} = \frac{1}{\det A} \sum_k \delta{ik} C_{kj} ? No: matrix form verifies \( A \adj A = (\det A) I \).
Example (2×2 Cofactors): For above \( A \), \( M_{11} = [a_{22}] \), \( C_{11} = (-1)^{1+1} a_{22} = a_{22} \); \( M_{12} = [a_{21}] \), \( C_{12} = (-1)^{1+2} a_{21} = -a_{21} \); similarly \( C_{21} = -a_{12} \), \( C_{22} = a_{11} \). Thus \( \adj A = \begin{pmatrix} C_{11} & C_{21} \ C_{12} & C_{22} \end{pmatrix} = \begin{pmatrix} a_{22} & -a_{12} \ -a_{21} & a_{11} \end{pmatrix} \).
- Cramer’s Rule (for \( \sum_j a_{ij} x_j = b_i \), \( i=1^n \), \( \det A \neq 0 \)): \( x_r = \frac{\det A_r}{\det A} \), where \( A_r \) replaces column \( r \) of \( A \) with \( [b_i]{i=1}^n \), so \( (A_r){ij} = a_{ij} \) if \( j \neq r \), else \( b_i \).
3. Linear Systems & Gaussian Elimination
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Augmented Matrix: \( [A b] = [a_{ij} b_i] \) for \( i=1^m \), \( j=1^n \). - Row Reduction to REF: Apply elementary ops (swap rows \( p \leftrightarrow q \); scale row \( p \) by \( c \neq 0 \): row \( p \leftarrow c \) row \( p \); add \( c \) row \( q \) to row \( p \)) to get row echelon form: leading entry (pivot) in row \( i \) at column \( p_i \geq p_{i-1} \), zeros below pivots.
- To RREF: Continue to zeros above pivots, scale pivots to 1.
- Rank: Number of nonzero rows in REF (or pivots).
- Solutions:
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Unique if rank \( A = n \), rank \( [A b] = n \) (nullity 0). -
Infinite if rank \( A = \) rank \( [A b] = r < n \) (n-r free vars). -
Inconsistent if rank \( A < \) rank \( [A b] \).
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- General Solution: \( x = x_p + x_h \), particular \( x_p \) from RREF, homogeneous \( x_h \) spans nullspace (free vars basis).
- Step Example (2×2 System Annotation): Solve \( a_{11} x_1 + a_{12} x_2 = b_1 \), \( a_{21} x_1 + a_{22} x_2 = b_2 \). Row2 ← Row2 - (a_{21}/a_{11}) Row1: new row2 = [0, a_{22} - (a_{21} a_{12}/a_{11}), b_2 - (a_{21} b_1 / a_{11}) ]. Back-sub: \( x_2 = \) … / det term, etc.
4. Vector Spaces
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Subspaces: Col(A) = span{ col j of A, j=1^n } = { \( \sum_j x_j \) col j x }; dim = rank A. - Row Space: Row(A) = Col(A^T); dim = rank A.
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Nullspace: Nul(A) = { x \( \sum_j a_{ij} x_j = 0 \) ∀ i }; basis from RREF free columns. - Rank-Nullity: rank A + dim Nul(A) = n.
5. Linear Transformations
- Matrix Rep: T(x)i = \( \sum_j a{ij} x_j \).
- Kernel: Ker T = Nul(A); Im T = Col(A).
6. Eigenvalues
- Char Poly: det(A - λ I) = 0, where (A - λ I){ij} = a{ij} - λ δ_{ij}.
- Eigenvectors: For λ, solve \( \sum_j (a_{ij} - λ δ_{ij}) v_j = 0 \), v = [v_j] ≠ 0.
- Diagonalizable: If alg mult(λ_k) = geo mult(λ_k) ∀ k (full eigenspace dim), then A = P D P^{-1}, D_{ij} = λ_i δ_{ij}, P columns = eigenvectors.
- Symmetric: A = A^T ⇒ orthogonal diag: A = Q D Q^T, Q^T Q = I.
7. Inner Products & Orthogonalization
- Euclidean: <u,v> = \( \sum_i u_i v_i = u^T v \).
- Gram-Schmidt Step: For basis {v^{(k)}}, u^{(1)} = v^{(1)}; u^{(k)} = v^{(k)} - \sum_{m=1}^{k-1} \proj_{u^{(m)}} v^{(k)}, where \proj_w v = \frac{<v,w>}{<w,w>} w (component \( \sum_i … \)).
| Quick Verification Tip: Always row-reduce [A - λI | 0] for eigenspaces; check det via expansion if n≤3. |
Linear Algebra Cheat Sheet - Indexed Formulas
Self-Exam 02198 Review