Vectors: Basic Operations
This blog is based on Jong-han Kim’s Linear Algebra
Block Vectors
\[\mathbf{a} = \begin{bmatrix} b \\ c \\ d \end{bmatrix}\]Zero, ones, and unit vectors
n-vector모든 값이 $0$ 이면, $0_n$, $0$라 표현한다.n-vector모든 값이 $1$ 이면, $\mathbf{1}_n$, $\mathbf{1}$라 표현한다.unit vector는 하나의 값이 $1$, 나머지는 $0$으로 채워짐.
e.g., 길이가 $3$인 단위벡터
\[\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \quad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \quad \mathbf{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\]Sparsity
벡터에 $0$으로 채워진 경우가 많다. 이를 효율적으로 계산할 수 있음.
$\mathbf{mnz}(x)$: non-zero인 값의 수
e.g., zero vector, unit vector
Properties of vector addition
commutative: $a + b = b + a$associative: $(a + b) + c = a + (b + c)$
Scaler-vector multiplication
scalar $\beta$, n-vector $\mathbf{a}$
\[\beta \mathbf{a} = (\beta \mathbf{a}_1, \dots, \beta \mathbf{a}_n)\]e.g.,
\[(-2) \begin{bmatrix} 1 \\ 9 \\ 6 \end{bmatrix} = \begin{bmatrix} -2 \\ -18 \\ -12 \end{bmatrix}\]Properties of scalar-vector multiplication
associative: $(\beta \gamma)\mathbf{a} = \beta(\gamma \mathbf{a})$left distributive: $(\beta + \gamma) \mathbf{a} = \beta\mathbf{a} + \gamma\mathbf{a}$right distributive: $\beta(\mathbf{a} + \mathbf{b}) = \beta\mathbf{a} + \beta\mathbf{b}$
Linear combinations
vectors $\mathbf{a}_1, \dots, \mathbf{a}_m$, scalars $\beta_1, \dots,\beta_m$ 의 linear combination은 $\beta_1\mathbf{a}_1 + \dots + \beta_m\mathbf{a}_m$ 이다.
e.g., for any n-vector $\mathbf{b}$
\[\mathbf{b} = \mathbf{b}_1\mathbf{e}_1 + \dots + \mathbf{b}_n\mathbf{e}_n\]Linear product
Inner product (or dot product) of n-vectors $\mathbf{a}$ and $\mathbf{b}$ is
\[\mathbf{a}^T\mathbf{b} = \mathbf{a}_1\mathbf{b}_1 + \dots + \mathbf{a}_n\mathbf{b}_n\]다른 notation: $<a, b>,\ <a \vert b>,\ (a, b),\ a \cdot b$
e.g.,
\[\begin{bmatrix} -1 \\2 \\2 \end{bmatrix}^T \begin{bmatrix} 1 \\0 \\-3 \end{bmatrix} = (-1)(1) + (2)(0) + (2)(-3) = -7\]Properties of inner product
- $\mathbf{a}^T\mathbf{b} = \mathbf{b}^T\mathbf{a}$
- $(\gamma\mathbf{a})^T\mathbf{b} = \gamma(\mathbf{a}^T\mathbf{b})$
- $(\mathbf{a}+\mathbf{b})^T\mathbf{c} = \mathbf{a}^T\mathbf{c}+\mathbf{b}^T\mathbf{c}$
e.g.,
\[(\mathbf{a}+\mathbf{b})^T (\mathbf{c}+\mathbf{d}) = \mathbf{a}^T\mathbf{c}+\mathbf{a}^T\mathbf{d}+\mathbf{b}^T\mathbf{c}+\mathbf{b}^T\mathbf{d}\]중요 예시
행렬에서 자주 사용하는 수식이다.
\[\mathbf{e}^T_i\mathbf{a} = \mathbf{a}_i \quad \text{(picks out ith entry)}\] \[\mathbf{1}^T\mathbf{a} = \mathbf{a}_1 + \dots + \mathbf{a}_n \quad \text{(sum of entries)}\] \[\mathbf{a}^T\mathbf{a} = \mathbf{a}^2_i + \dots + \mathbf{a}^2_n \quad \text{(sum of squares of enties)}\]