Schedule

• Linear independence
• Spanning a space
• Basis and Dimension
该章节中说到的无关性和张成空间均指的是向量组而非矩阵.

Independence – 线性无关性

Suppose A is m\times n with m<n, then there are nonzero solutions to Ax=0 (more unknowns than equations).
The reason why there is solution is there will be free variables.

When vectors x_1,x_2,…x_n are independent?

c_1x_1 + c_2x_2 + … + c_nx_n \ne 0 除非 all\phantom{1}c_i=0

When v_1,v_2,…v_n are columns of A:

• They are independent if the null-space of A is only zero vector \rightarrow (r=n).
• Then are dependent if Ac=0 for some non-zero c \rightarrow (r<n).

Spanning – 张成

Vectors v_1,…v_l span a space means the space consists of all combinations of those vectors.

Basis for a space is a sequence of vectors v_1,v_2,…v_d that has two properties:

• they are independent.
• they span the space.

Example:
Space is R^3.

For R^n, n vectors give basis if the n\times n matrix with those columns is invertible.

Given a space, every basis for the space has the same number of vectors and this number is the dimension of the space.

Independence, that looks at combinations not being zero.
Spanning, that looks at all the combinations.
Basis, that’s the one that combines independences and spanning.
Dimension, the number of vectors in any basis.

Example:
Space is C(A) = \begin{bmatrix}1&2&3&1\1&1&2&1\1&2&3&1\end{bmatrix}
2 = rank(A) = # pivot columns = dimension of C(A)

dimC(A) = r
dimN(A) = n-r = \text{# free variables}