A.1 · พีชคณิตเชิงเส้น
GRE Subject Test · GRE Mathematics หัวข้อ 3 3:06 การบรรยายภาษาอังกฤษ · คำบรรยายภาษาอังกฤษ + 中文 ลอยตัวบนภาพ
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Transcript
This unit relates rank, nullity and the solutions of linear systems; analyses vector spaces and linear maps; and computes eigenvalues, determinants and the conditions for diagonalisation.
本单元把秩、零化度与线性方程组的解联系起来；分析向量空间和线性映射；并计算特征值、行列式以及可对角化的条件。
A basis is an independent spanning list, so every vector has unique coordinates.
基是线性无关的生成组，所以每个向量的坐标唯一。
To test independence of columns, solve A c equals 0: only the zero solution means independent.
检验列向量是否无关，就解 Ac = 0：只有零解才说明无关。
A spanning set can carry redundant vectors, so spanning alone is not a basis.
生成集可能含有多余向量，所以只生成不等于是基。
Row reduction exposes pivots and free variables.
行化简揭示主元和自由变量。
Rank is the dimension of the image, the number of pivots; nullity is the dimension of the kernel.
秩是像的维数，即主元个数；零化度是核的维数。
For T from R 3, rank is one and nullity two: together they make the domain dimension, 3.
对从 R³ 出发的 T，秩为 1，零化度为 2：加起来等于定义域的维数 3。
A zero row with a nonzero right side means no solution.
系数为零的行对应非零右端，说明无解。
A square matrix is invertible exactly when its determinant is nonzero, its kernel is zero and its rank is full.
方阵可逆，当且仅当行列式非零、核为零、秩为满秩。
None of this needs every entry to be nonzero.
这些都不要求每个元素非零。
A row swap flips the determinant's sign; adding a multiple of a row leaves it unchanged.
交换两行改变行列式的符号；把一行的倍数加到另一行则不改变它。
Eigenvalues are the roots of the determinant of A minus lambda I.
特征值是 det(A − λI) 的根。
Diagonalising needs a full basis of eigenvectors.
对角化需要由特征向量组成的完整基。
Distinct eigenvalues guarantee one; a repeated eigenvalue may have too small an eigenspace.
互不相同的特征值保证这一点；重根可能对应的特征空间太小。
A real symmetric matrix always has an orthonormal eigenbasis.
实对称矩阵总有标准正交的特征基。
The shear slides the unit square sideways.
剪切把单位正方形横向推移。
Its characteristic polynomial is 1 minus lambda, squared, and eigenvectors need y equals 0: only the x-direction stays put, while every other vector turns.
它的特征多项式是 (1 − λ)²，特征向量需满足 y = 0：只有 x 方向保持不动，其他每个向量都转了方向。
One independent eigenvector cannot diagonalise a two-by-two matrix.
一个无关的特征向量不能使二阶矩阵对角化。
The algebraic multiplicity of an eigenvalue is not automatically the dimension of its eigenspace.
特征值的代数重数不一定等于其特征空间的维数。
The second row is twice the first, so the rank is one and every output is s, 2s.
第二行是第一行的两倍，所以秩为 1，每个输出都是 (s, 2s)。
The kernel, x equals minus 2y minus z, has two independent vectors.
核 x = −2y − z 有两个无关向量。
So 3, 6 has a two-parameter family of solutions, and 3, 7 has none, because 7 is not twice 3.
所以 (3, 6) 有一个双参数的解族，而 (3, 7) 无解，因为 7 不是 3 的两倍。
The characteristic polynomial is 2 minus lambda, squared, and the determinant is 4.
特征多项式是 (2 − λ)²，行列式是 4。
The eigenspace has dimension one over either field: invertible, but not diagonalisable.
在任一数域上特征空间维数都是 1：可逆，但不可对角化。
Write A as 2I plus N with N squared zero; the binomial expansion leaves two terms.
把 A 写成 2I + N，其中 N² = 0；二项式展开只剩两项。
Check yourself.
自我检测。
A spanning set is a basis only if independent.
生成集只有在线性无关时才是基。
Rank plus nullity is the domain dimension.
秩加零化度等于定义域维数。
Invertible means a nonzero determinant.
可逆意味着行列式非零。
A repeated eigenvalue need not allow diagonalisation.
重特征值不一定能对角化。
And a real symmetric matrix has an orthonormal eigenbasis.
实对称矩阵有标准正交特征基。