Question

What are the coordinates of (3 comma space 8 )relative to the basis open curly brackets space (1 comma space 1 )comma space (0 comma space 1 )space close curly brackets

Answer 1

Answer:

(3,5)

Explanation:

Given the basis β={(1,1),(0,1)} and x=(3,8), I am to find the corresponding coordinate vector [x]β. I claim that the coordinate vectors entries $x_{1}$ and $x_{2}$ meet the following criterion:

$x_{1} (1,1) +x_{2} (0,1)=(3,8)$

Placing this in matrix form

$\left(\begin{array}{cc}1 & 0\\ 1 & 1 \end{array} \right)\left(\begin{array}{c}x_{1} \\ x_{2}\end{array} \right)$

$x_{1} +0x_{2} =3$

$x_{1} =3$

Also,

$1x_{1} +1x_{2} =8$

$3 + 1x_{2} =8$

$x_{2} =8-3=5$

The coordinate matrix therefore is given as (3,5)