Lecture 001

$\begin{cases} 2x + y = 5 \\ 4x + 3y = 11 \\ \end{cases}$

Solution:

$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$

The raw picture: a picture of the line intersection created by two lines

You can also write the above equation in the vector way:

$x \begin{pmatrix} 2 \\ 4 \\ \end{pmatrix} + y \begin{pmatrix} 1 \\ 3 \\ \end{pmatrix} = \begin{pmatrix} 5 \\ 11 \\ \end{pmatrix}$

You can think of: find the constant scaling factor $x$ and $y$ so that the addition of two vector addition is the result $\begin{pmatrix} 5\\11 \end{pmatrix}$ (linear combination)

Whether there are Solutions

No Solution:

Raw Picture: parallel line
Vector Space: same direction

One Solution:

Raw Picture: intersection
Vector Space: vector addition

Infinite Solution:

Raw Picture: same line
Vector Space: three vectors are at the same direction
If infinite many solutions: let $x$ be an arbitrary value $p$ , then $y = 3-2p$ .

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