جبر خطیLinear Algebra
بردارها، ماتریسها، مقادیر ویژه و فضاهای برداری.
کنکور ارشد علوم کامپیوتر
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کارتها · 19
The bottom-right block H of M^(-1) satisfies det(M)det(H)=det(A), the diagonally opposite block.
A map between C-vector spaces is C-linear iff it is additive and homogeneous for every complex scalar.
Eigenvalues are the roots of det(A-λ I) , and structure pins them down: triangular matrices show them on the diagonal, nilpotents force 0 , and real symmetric matrices have real eigenvalues with orthogonal eigenvectors.
An invertible idempotent matrix must equal the identity, so all its eigenvalues are 1.
An overdetermined system Ax=b is fit by least squares via the normal equations A^T A x = A^T b.
For an m x n matrix, dim(null space) = n - rank(A), so any wide matrix (more columns than rows) always has nonzero kernel.
Every square matrix satisfies its own characteristic polynomial; rearranging that polynomial expresses A^(-1) as a cheap polynomial in A — no cofactor matrix needed.
The determinant is linear in each row and alternating, so scaling a row scales it, swapping rows flips the sign, and any repeated row makes it zero — that is all you need to evaluate "transformed-matrix" determinant problems.
For any square matrix the trace equals the sum of the eigenvalues and the determinant equals their product — and these are exactly the constraints that pin down unknown eigenvalues on the exam.
For two subspaces, dim(U+W)=dim U+dim W-dim(U∩ W) — an inclusion–exclusion for dimensions that pins down (or bounds) the size of an intersection.
A linear map becomes a matrix once you fix a basis: its j -th column is the image of the j -th basis vector, written in that same basis's coordinates.
An R-linear map on C need not be C-linear; realify it as a 2x2 real matrix in the basis 1, i and take that matrix's determinant.
Given an ordered basis B of a vector space, every vector has a unique tuple of coordinates: the scalars in its expansion over B. Finding them means solving a linear system.
Factor out a common matrix power, then left- or right-multiply by its inverse to cancel it — only legal when that factor is invertible.
The minimal polynomial equals the characteristic polynomial exactly when each eigenvalue has a single Jordan block (one eigenvector apiece).
When an unknown matrix appears together with its transpose and trace, write it with literal entries, expand both sides entrywise, and turn the matrix identity into a small linear system.
A system Ax=b is ill-conditioned when a tiny change in the data ( b or A ) produces a huge change in the solution x . The condition number κ(A)=lVert ArVert lVert A^(-1)rVert≥ 1 measures this sensitivity: it is the worst-case amplification factor from relative input error to relative output error.
Matrix similarity ( A=PBP^(-1) ) survives both inverting and transposing: if A~ B then A^(-1)~ B^(-1) and A^t~ B^t . Plus every square matrix is similar to its own transpose, A~ A^t . Chaining these rules lets you transform a given similarity into many others.
A square system Ax=b has exactly one solution for every right-hand side if and only if A is invertible, i.e. det(A)≠ 0 — so "for which parameter is the solution unique?" always reduces to solving det(A)≠ 0 .