# 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

• If infinite many solutions: let $x$ be an arbitrary value $p$, then $y = 3-2p$.