[ 机器学习 - 吴恩达 ] Linear Algebra review | 3-3 Matrix-vector multiplication

Example - 1

\[\begin{bmatrix} 1&3\\ 4&0\\ 2&1\\ \end{bmatrix}\begin{bmatrix} 1\\ 5\\ \end{bmatrix}=\begin{bmatrix} 16\\ 4\\ 7\\ \end{bmatrix}\]

\[\quad\ 3\times2\quad 2\times1\quad\ 3\times1\ \ \]

\(1\times1+3\times5=16\)
\(4\times1+0\times5=4\)
\(2\times1+1\times5=7\)

Details - 1

\(A\quad\quad\quad\quad\quad \times \quad \quad x\quad \quad =\quad\quad\quad\quad\quad y\)
\(m\times n\ matrix\\(m\ rows,\\\ n\ columns)\) \(\quad n\times1\ matrix\\(n-dimensional\\ \quad \quad vector)\) \(m-dimensional\\ \quad \quad vector\)

To get \(y_i\), multiply \(A's\ i^{th}\) row with elements of vector \(x\), and add them up.

Example - 2

\[\begin{bmatrix} 1&2&1&5\\ 0&3&0&4\\ -1&-2&0&0\\ \end{bmatrix}\begin{bmatrix} 1\\ 3\\ 2\\ 1\\ \end{bmatrix}=\begin{bmatrix} 14\\ 13\\ -7 \end{bmatrix}\]

House sizes

\(\begin{matrix} 2104\\ 1416\\ 1534\\ 852\\ \end{matrix}\)??????\(h_\theta(x)=-40+0.25x\)
构建样本矩阵并与参数向量相乘，即可得到所有的预测值：

\[\begin{bmatrix} 1&2104\\ 1&1416\\ 1&1534\\ 1&852\\ \end{bmatrix}\times \begin{bmatrix} -40\\ 0.25\\ \end{bmatrix} = \begin{bmatrix} h_\theta(2104)\\ .\\ .\\ .\\ \end{bmatrix}\]

Prediction (4x1) = DataMatrix (4x2) x Parameters (2x1)
明显快于利用for循环遍历样本进行预测，并且代码得到了简化