vector-spaces.wiki (1625B)
1 == Vector spaces == 2 subspace of $\Re^n$ is any set H in $\Re^n$ that has properties: 3 a) The zero vector is in H 4 b) For each u and v in H, the sum $u + v$ is in H 5 c) For each u in H and each scalar c, the vector $cu$ is in H 6 7 the zero subspace is the set that only contains zero vector in $\Re^n$ 8 9 === Column space and null space of a matrix === 10 column space: set of all linear combinations of the columns of a matrix. it's the _span_ of the columns of the matrix. 11 12 column space of m × n matrix is subspace of $\Re^m$ 13 14 null space: set of all solutions of equation $Ax = 0$. 15 16 null space of an m × n matrix is subspace of $\Re^n$. 17 18 to determine if p is in Nul A, check if Ap = 0. if so, p is in Nul A. 19 20 === Basis for a subspace === 21 basis for subspace H of $\Re^n$ is linearly independent set in H spanning H 22 23 the pivot columns of a matrix form the basis for its column space. 24 25 === Coordinates === 26 let $H \in \Re^n$ be subspace with $B = \{ b_1, \dots, b_p\}$. then for all x ∈ H, there are unique $c_1, \dots, c_p$ such that $x = c_1 b_2 + \dots + c_p b_p$. (to prove this theorem, use a contradiction on uniqueness) 27 28 the coordinates of x w.r.t. B are $c_1, \dots, c_p$. 29 30 the coordinate system of x w.r.t. B is $[x]_B = \begin{bmatrix}c_1\\ \dots\\ c_p\end{bmatrix}$ 31 32 === Dimension of a subspace === 33 let $H \in \Re^n$ be a subspace with basis $B=\{ b_1, \dots, b_p \}$. then every basis for H comprises p vectors. 34 35 the dimension of H is the number of basis vectors in _any_ basis for H. 36 37 dim Col A = #pivot columns (rank A) 38 39 dim Nul A = #free variables in Ax = 0 40 41 Rank theorem: dim Col A + dim Nul A = #columns 42