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

(3,5)

Step-by-step 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 [TeX] x_{1} [/TeX] and [TeX] x_{2} [/TeX] meet the following criterion:

[TeX] x_{1} (1,1) +x_{2} (0,1)=(3,8)[/TeX]

Placing this in matrix form

[TeX]\left(\begin{array}{cc}1 & 0\\ 1 & 1 \end{array} \right)\left(\begin{array}{c}x_{1} \\ x_{2}\end{array} \right) [/TeX]

[TeX] x_{1} +0x_{2} =3[/TeX]

[TeX] x_{1} =3[/TeX]

Also,

[TeX] 1x_{1} +1x_{2} =8[/TeX]

[TeX] 3 + 1x_{2} =8[/TeX]

[TeX] x_{2} =8-3=5[/TeX]

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