### Span The $\text{Span}(f_{1},\dots, f_{n})$ is the set of all functions $c_{1}f_{1}+\dots +c_{n}f_{n}$ where $c_{1},\dots,c_{n}$ are real numbers ### Vector Space Then $S$ is a real vector space if all of the following are true: 1. There is a zero vector in $S$ There is a vector $\mathbf{0}$ in $S$ such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for every vector $\mathbf{v}$ in $S$. 2. Closed under scalar multiplication Multiplying any one vector in $S$ by a real number gives another vector in $S$ 3. Closed under vector addition Adding any two vectors in $S$ gives another vector in $S$ ### Dimension The dimension of a vector space is the number of vectors in any basis. The dimension of the space of solutions to an $n^\text{th}$ order homogeneous ODE with constant coefficients is $n$. ### Basis A basis of a vector space $S$ is a list of vectors $v_1, v_2, …, v_n$, such that 1. $\text{Span}(v_1, v_2, …, v_n) = S$ 2. The vectors $v_1, v_2, …, v_n$ are linearly independent. A set of vectors $\{ v_{1}, v_{2}, \dots , v_{k}\}$ **generate** subspace $S$ if each $v_{i}$ is in $S$ and $\text{span}(\{ v_{1}, v_{2}, \dots , v_{k}\}) = S$. ### Orthonormal Bases A set of vectors $\{ v_{1}, v_{2}, \dots , v_{k}\}$ is an orthonormal basis for the subspace $S$ if they generate $S$, all have length one and are pairwise orthogonal. ### Orthogonal Decomposition Every vector can be written as $w = u + v$, where $\begin{cases}u \in U \\ v \in U^{\perp}\end{cases}$ and $u \cdot v=0$ ### Columnspace The columnspace (CA) of an $m$ x $n$ matrix $A$ is the span of its $n$ columns. $Ax=b$ has a solution if and only if $b \in C(A)$ If $A$ is an $n$ x $n$ matrix, $A$ is invertible if and only if $C(A)=\mathbb{R}^{n}$ *Proof:* - If$A$ is invertible, for all $b\in \mathbb{R}^{n}$, $Ax=b$ for $x=A^{-1}b$* - If $C(A)=\mathbb{R}^{n}$, there exists $Ax_{1}=e_{1}$ for each of the standard basis vectors $e_{1},e_{2}, \dots e_{n}$ such that $A\begin{bmatrix}x_{1} & x_{2} & \dots x_{n}\end{bmatrix}= I$, which gives us $\begin{bmatrix}x_{1} & x_{2} & \dots x_{n}\end{bmatrix} = A^{-1}$ ### Nullspace The nullspace is all $x$ such that $Ax = 0$ If $A$ is $n$ x $n$ then $C(A) = \mathbb{R}^{n}$ if and only if $N(A) = 0$ *Proof:* - If $N(A)\neq 0$ then we must get a row of all zeros in the row echelon form of A, making it singular and so $C(A) \neq \mathbb{R}^{n}$ - If $N(A)=0$, the row echelon form of $A$ must have all nonzero pivots, making A invertible so that $C(A)=\mathbb{R}^{n}$