Linear Algebra

Lecture #1

W. Gilbert Strang

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n linear equations, n unknowns

• row picture
• col picture
• Matrix form

$$\{\begin{array}{c}2x-y=0\\ -x+2y=3\end{array}$$

1 Row Picture

2 Column Picture

What are all combinations ? The whole plane.

3 Matrix form

matrix form:
$$
\begin{bmatrix}
2 & -1 \
-1 & 2 \
\end{bmatrix}

\begin{bmatrix}
x \
y \
\end{bmatrix}

\begin{bmatrix}
0 \
3 \
\end{bmatrix}
$$

$AX=b$

4 3 unknowns and 3 equations

5 Can I solve $Ax=b$ for every b ?

Do the linear combinations of the columns fill three dimentional space ?

A non-singular matrix or an invertible matrix.

非奇异矩阵 (又称 可逆矩阵 或 正则矩阵) 是一种存在逆元的方块矩阵。相反的，若方阵不存在逆元，则称为 奇异矩阵。

$$Ax=b$$

$$
\begin{bmatrix}
2 & 5 \
1 & 3 \
\end{bmatrix}
\begin{bmatrix}
1 \
2
\end{bmatrix}

1
\begin{bmatrix}
2 \
1
\end{bmatrix}
+
2
\begin{bmatrix}
5 \
3
\end{bmatrix}

\begin{bmatrix}
12 \
7
\end{bmatrix}
$$