Đại số tuyến tính
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| rank/ræŋk/ | Bậc (Rank) |
| nullity/ˈnʌlɪti/ | The dimension of the null space of a matrix or linear map. |
| eigenspace/ˈaɪdʒənspeɪs/ | The set of vectors v with Av = λv for one eigenvalue λ, including zero. |
Nonzero entries, zero determinant
- $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ has no zero entries, yet its second row is twice the first: it is singular.
- GRE linear algebra tests rank 秩, kernels, determinants and eigenvectors, and the conditions linking them.
Independence, span, basis
- Columns of $A$ are independent exactly when $A\mathbf{c} = \mathbf{0}$ has only $\mathbf{c} = \mathbf{0}$.
- A basis is an independent spanning list; each vector has unique coordinates. A spanning set may contain redundant vectors.
Row reduction and rank–nullity 零度
- Row reduction keeps the solution set. Rank = number of pivots = dimension of the column space.
- For $A$ with $n$ columns: $\operatorname{rank} A + \operatorname{nullity} A = n$.
- A row $[0 \; \cdots \; 0 \mid b]$ with $b \neq 0$ means no solution; free variables give infinitely many only once the system is consistent.
A $4 \times 5$ matrix has rank 3. What is the dimension of its null space?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Rank–nullity: rank + nullity = number of columns. With 5 columns and rank 3, the null space has dimension 5 − 3 = 2.
Determinants and invertibility
- For square $A$: invertible $\iff \det A \neq 0 \iff \ker A = \{\mathbf 0\} \iff$ rank $= n$.
- Triangular: $\det$ is the product of the diagonal. A row swap flips the sign; adding a multiple of one row to another changes nothing; scaling a row scales $\det$.
Find $\det \begin{pmatrix} 2 & 1 \\ 4 & 5 \end{pmatrix}$.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
For a 2 × 2 matrix, det = ad − bc = 2 · 5 − 1 · 4 = 10 − 4 = 6.
If $A$ is square and $\det A = 0$, then
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Zero determinant means a nontrivial kernel.
Eigenvalues and diagonalisation
- Eigenvalues are roots of $\det(A - \lambda I)$. Diagonalisable means a full basis of eigenvectors.
- Distinct eigenvalues guarantee it; a repeated eigenvalue may not have enough eigenvectors. Real symmetric matrices always diagonalise orthogonally.
$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$: $\lambda = 1$ twice, but $(A - I)\mathbf v = \mathbf 0$ forces $v_2 = 0$, a one-dimensional eigenspace 特征空间. $A$ is not diagonalisable; the identity, with the same eigenvalues, is.
Diagonalisable over ℝ?
Decide diagonalisability from eigenvalues and eigenspaces.
Which matrix is NOT diagonalisable over the reals?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Eigenvalue 2 twice with a one-dimensional eigenspace.
The dimension of an eigenspace always equals the algebraic multiplicity of its eigenvalue.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
False. Geometric multiplicity (eigenspace dimension) is at least 1 and at most the algebraic multiplicity, and can be smaller: the matrix [[1, 1], [0, 1]] has eigenvalue 1 twice but a one-dimensional eigenspace.
Recap
- Rank + nullity = number of columns.
- $\det A = 0 \iff$ nontrivial kernel; entries being nonzero is irrelevant.
- Algebraic multiplicity can exceed eigenspace dimension; then diagonalisation fails.
- eigenspace: The set of vectors v with Av = λv for one eigenvalue λ, including zero.
- nullity: The dimension of the null space of a matrix or linear map.