vector-spaces.md (1654B)
1 +++ 2 title = 'Vector spaces' 3 template = 'page-math.html' 4 +++ 5 6 # Vector spaces 7 subspace of $\Re^n$ is any set H in $\Re^n$ that has properties: 8 a) The zero vector is in H 9 b) For each u and v in H, the sum $u + v$ is in H 10 c) For each u in H and each scalar c, the vector $cu$ is in H 11 12 the zero subspace is the set that only contains zero vector in $\Re^n$ 13 14 ## Column space and null space of a matrix 15 column space: set of all linear combinations of the columns of a matrix. it's the _span_ of the columns of the matrix. 16 17 column space of m × n matrix is subspace of $\Re^m$ 18 19 null space: set of all solutions of equation $Ax = 0$. 20 21 null space of an m × n matrix is subspace of $\Re^n$. 22 23 to determine if p is in Nul A, check if Ap = 0. if so, p is in Nul A. 24 25 ## Basis for a subspace 26 basis for subspace H of $\Re^n$ is linearly independent set in H spanning H 27 28 the pivot columns of a matrix form the basis for its column space. 29 30 ## Coordinates 31 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) 32 33 the coordinates of x w.r.t. B are $c_1, \dots, c_p$. 34 35 the coordinate system of x w.r.t. B is $[x]_B = \begin{bmatrix}c_1\\\\ \dots\\\\ c_p\end{bmatrix}$ 36 37 ## Dimension of a subspace 38 let $H \in \Re^n$ be a subspace with basis $B=\{ b_1, \dots, b_p \}$. then every basis for H comprises p vectors. 39 40 the dimension of H is the number of basis vectors in _any_ basis for H. 41 42 dim Col A = #pivot columns (rank A) 43 44 dim Nul A = #free variables in Ax = 0 45 46 Rank theorem: dim Col A + dim Nul A = #columns 47