maths.free › Linear Algebra › Vector spaces, span and linear independence
Vector spaces, span and linear independence
The axioms, spanning sets, independence, basis and dimension.
A vector space is a set closed under addition and scaling. Vectors are independent when no combination cancels; a basis is an independent spanning set, and every basis has the same size: the dimension. Picture it: two vectors spanning a plane unless they are parallel, the 3D view of the column vectors. Think it: rank is the dimension of the column space; rank + nullity = number of columns.
مثال کار شده · rank of [[1,2],[2,4]]
قدم به قدم
- \left[\begin{matrix}1 & 2\\2 & 4\end{matrix}\right]
Row-reduce; the rank is the number of non-zero rows (pivots).
- \left[\begin{matrix}1 & 2\\0 & 0\end{matrix}\right]
R2 ← R2 − (2)·R1 to clear column 1.
- \operatorname{rank} A = 1
جواب رو نشون بده
حالا تو يه مساله انتخاب کن، يا خودت رو بنويس يا بکشي هر گام، يه تصوير، جواب مخفي ميشه تا وقتي که تو بپرسي
نمادهای استفاده‌شده در اینجا
هر نماد را برای تعریف کامل، تصویر و معنی هر حرف در آن بزنید.
چطور: Vector spaces, span and linear independence
- Put the vectors as columns of a matrix and row-reduce.
- Pivot columns are a basis of the span; the number of pivots is the rank.
- Free columns give the null space: the dependencies among the vectors.
سوالاتي که مردم ميپرسن
What does a determinant mean geometrically?
It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.
What is an eigenvector?
A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.
Why is matrix multiplication not commutative?
Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.
بیشتر در Linear Algebra
DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationRow reductionOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis