1. Span Verb : A Set of vectors is said to span a space if the set of all their linear combinations is the space Noun : $span(v_{1}, v_{2}) = \{ c_{1} v_{1} + c_{2} v_{2}, \forall c_{1}, c_{2} \in \mathbb{R} \}$ Note : vectors that span a space are not necessarily independent Fact : columns of $A$ span $C(A)$, special solutions of $A$ span $N(A)$ 2. Basis A Basis of a space is a set of linearly ..
1. Rank The rank of matrix in the number of pivots ($r(A) = \# $ of pivots) Also defiend as the maximum number of linearly independent columns (will be discussed later) The rank of a matrix gives in a sense the true size of the matrix. Consider an $(m \times n)$ matrix $A$ whose rank is few than $m$. While $A \mathbf{x} = \mathbf{0}$ seems to have $m$ linear equations, $m-r$ out of $m$ equations..
1. Vector Spaces A set of vectors $V$ is called a vector space if it satisfies the following axioms $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}, \mathbf{v} \in V$ $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w}), \forall \mathbf{u} \mathbf{v} \mathbf{w}$ There exists a zero vector $\mathbf{0} \in V$, such that $\mathbf{v} + \mathbf{0} = \mathbf{v}, \forall \mathbf{v}..
1. Inverse Matrices Square matrix is said to be invertible if its inverse matrix exists. Sqaure matrix that is not invertible is singular. 2. Properties of Inverse Matrix P1. $A$ is invertible if and only if (iff) guassian elimination produces $n$ pivots ($A \mathbf{x} = \mathbf{b}$ 꼴에서 $A$의 열의 linear combination 관점에서 봤을 때 모든 $x_{i}$가 유일하게 존재한다는 말이므로 span위에 $\mathbf{b}$가 존재한다) P2. $A$가 invertibl..
1. Gaussian Elimination 아래와 같은 linear equations를 풀려고 한다. \begin{align} x_{1} + x_{2} &= 2 \\ 2x_{1} - x_{2} &= -4 \end{align} 일반적으로 두 등식 간에 연산을 취하여 아래의 꼴을 만들면 답을 쉽게 구할 수 있다. \begin{align} \\ x_{1} + x_{2} &= 2 \\ -3x_{2} &= -8 \end{align} 즉, 일단 바로 위의 등식과 같은 triangular form을 얻으면 풀렸다고 봐도 된다. 이렇게 행렬을 triangular form으로 만들기 위한 방법을 Gaussian Elimination Method라고 한다. 즉, $A \mathbf{x} = \mathbf{b}$에서 $A$..
1. 행렬과 벡터의 곱의 계산적 의미 \begin{align*} A{x}= \left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n}\\ a_{21} & a_{22} & \ldots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{array} \right] \left[ \begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n \end{array} \right] = \left[ \begin{array}{c} a_{11}x_1+a_{12}x_2 + \cdots + a_{1n} x_n\\ a_{21}x_1+a_{22}x_2 + \cdots..
Linear Combination The set $\mathbb{R}^{n}$ or an infinite line can be represented as linear combinations of vectors Example 1 : $c\begin{bmatrix}1 \\ 0\end{bmatrix} + d\begin{bmatrix}0 \\ 1\end{bmatrix}$ spans $\mathbb{R}^2$ Example 2 : $c\begin{bmatrix}1 \\ 1\end{bmatrix} + d\begin{bmatrix}2 \\ 2\end{bmatrix}$ is an infinite line Lengths and Dot (Inner) Products Length : $\|v\| = \sqrt{v \cdot..