\(A\mathbf{x}\) : Product of A and x
The linear combination of the columns of \(A\) using the corresponding entries in \(\mathbf{X} \mbox{ as weights}\) → x 벡터 엔트리가 가중치
📖 Theorem 3.
\(A\) is \(m \times n\) matrix, with columns \(a_1, \cdots, a_n \mathbf{b} \mbox{ is in }\mathbb{R}^m\)
- Matrix Equation : \(A\mathbf{x} = \mathbf{b}\)
- Vector Equation : \(x_1\mathbf{a_1} + x_2\mathbf{a_2} + \cdots + x_p\mathbf{a_p} = \mathbb{b}\)
- Augmented Matrix : \(\left[ \mathbf{a_1}, \mathbf{a_2}, \cdots, \mathbf{a_n} \mathbf{b}\right]\)
Have the same solution set!
📖 Theorem 4.
The followings are all true of all false:
- For each b in R^m, Ax = b has a solution
- Each b in R^m is a linear combination of the columns of A
- The columns of A spanR^m
- A has a pivot position in every row
📖 Theorem 5.
If A is an m x n matrix, u and v are vectors in R^m, And c is a scalar, then;
A(u + v) = Au + Av;
A(cu) = c(Au)