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10.018 MSS – Linear Algebra (Wk 8-13) – Side 1
Systems of Linear Equations
- REF: Leading 1 in each row, zeros below pivots, each pivot right of above
- RREF: REF + zeros above pivots + pivot cols have only the leading 1
- Parametric last row \([0\;0\;f(a)\;|\;g(a)]\): No soln if \(f(a)=0, g(a)\neq0\); Unique if \(f(a)\neq0\); Inf. if \(f(a)=0, g(a)=0\)
- Homogeneous \(A\mathbf{x}=\mathbf{0}\): always consistent; if \(n>m\) ⟹ ∞ solutions
Matrix Operations & Inverses
- \((AB)^T = B^T A^T\); \((AB)^{-1} = B^{-1}A^{-1}\)
- 2×2 inverse: \(\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\)
- Gauss-Jordan: \([A|I] \to [I|A^{-1}]\)
- Elementary matrices: Swap (self-inv), Scale \(k\) (inv: \(1/k\)), Add \(kR_j\to R_i\) (inv: subtract)
- \(A = E_1^{-1}\cdots E_k^{-1}\) if \(E_k\cdots E_1 A = I\)
Determinants & Trace
- Cofactor: \(C_{ij}=(-1)^{i+j}\det(M_{ij})\)
- Via row reduction: \(\det = \frac{\text{prod of REF diag}}{(-1)^{\text{swaps}}\times\text{prod of scale constants}}\)
- \(\det(AB)=\det A\det B\); \(\det(cA)=c^n\det A\); \(\det(A^{-1})=1/\det A\); \(\det(A^T)=\det A\)
- Triangular \(\det\) = product of diagonal entries
- \(\text{tr}(A)=\sum a_{ii}\); \(\text{tr}(AB)=\text{tr}(BA)\); \(\text{tr}(A+B)=\text{tr}(A)+\text{tr}(B)\)
Subspaces, Span, Linear Independence
- Subspace test: contains \(\mathbf{0}\) + closed under \(+\) and scalar \(\times\)
- LI check: form matrix → row reduce → all columns pivot ⟹ LI
- \(\text{span}\{v_1,...,v_k\}\) = all linear combos; span is always a subspace
10.018 MSS – Linear Algebra (Wk 8-13) – Side 2
Basis, Rank, Null Space
- Basis = LI + spans the space; \(\dim\) = # basis vectors
- row(A) basis: nonzero rows of RREF(A)
- col(A) basis: pivot columns of ORIGINAL A
- null(A): solve \(A\mathbf{x}=\mathbf{0}\), parametrise free vars
- Rank-Nullity: \(\text{rank}(A)+\text{nullity}(A)=n\)
- Rank = # pivots = dim(row) = dim(col)
Change of Basis
- \([\mathbf{x}]_E = B[\mathbf{x}]_B\); \([\mathbf{x}]_B = B^{-1}[\mathbf{x}]_E\)
- Between bases: \([\mathbf{x}]_B = B^{-1}A[\mathbf{x}]_A\)
- Orthogonal basis coords: \(c_i = \frac{\mathbf{u}\cdot\mathbf{v}_i}{\mathbf{v}_i\cdot\mathbf{v}_i}\)
Eigenvalues & Diagonalisation
- \(A\mathbf{v}=\lambda\mathbf{v}\); solve \(\det(A-\lambda I)=0\)
- 2×2: \(\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0\)
- Eigenspace \(E_\lambda = \text{null}(A-\lambda I)\)
- alg mult ≥ geo mult ≥ 1
- Diagonalisable ⟺ geo = alg for all \(\lambda\)
- \(A=PDP^{-1}\); \(A^k=PD^kP^{-1}\)
- \(\det(A)=\prod\lambda_i\); \(\text{tr}(A)=\sum\lambda_i\)
- Markov steady state: eigenvector for \(\lambda=1\), normalise
Linear Transformations
- \(T\) linear: \(T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})\), \(T(c\mathbf{u})=cT(\mathbf{u})\); \(T(\mathbf{0})=\mathbf{0}\)
- Matrix rep: \(A=[T(\mathbf{e}_1)|\cdots|T(\mathbf{e}_n)]\)
- From pairs: \([T]=[w_1\cdots w_n][u_1\cdots u_n]^{-1}\)
- Composite \(S\circ T\): matrix = \(BA\)
Standard Transforms (ℝ²)
- Rot CCW θ: \(\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\) CW θ: \(\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}\)
- Reflect x-axis: \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\) y-axis: \(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\)
- Reflect y=x: \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\) y=−x: \(\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\)
- Reflect line at θ: \(\begin{bmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{bmatrix}\)
Projection & Least Squares
- Proj onto dir a: \(P=\frac{\mathbf{a}\mathbf{a}^T}{\mathbf{a}^T\mathbf{a}}\); \(P^2=P\)
- Distance: \(\|\mathbf{v}-P\mathbf{v}\|\)
- Least squares: \(\mathbf{c}=(A^TA)^{-1}A^T\mathbf{b}\)
Common Exam Steps
- Parametric system: row reduce → check last row condition
- Inverse: \([A|I]\to[I|A^{-1}]\)
- Det: row reduce (track ops) or cofactor expand along row/col with most 0s
- Eigenvalues: \(\det(A-\lambda I)=0\) → \(\text{null}(A-\lambda I)\) for each
- Diagonalise: check geo=alg; \(P\)=[eigenvectors], \(D\)=diag(eigenvalues)
- \(A^k\): diagonalise → \(PD^kP^{-1}\)