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## Homework Statement

Let E = {“ax+by+cz = d” | a; b; c; d ∈ R} be the set of linear equations

with real coefficients in the variables x, y and z. Equip E with the usual operations

on equations that you learned in high school. addition of equations, denoted below

by “⊕” and multiplication by scalars, denoted below by “[itex]\odot[/itex]”.

You may assume without proof that E is a vector space.

Find a spanning set for E.

(You must justify your answer.)

## Homework Equations

“ax+by +cz = d”⊕“ex+fy +gz = h” = “(a+e)x+(b+f)y +(c+g)z = d+h”

∀k ∈ R; k [itex]\odot[/itex] “ax + by + cz = d” = “ka x + kb y + kc z = k d”

## The Attempt at a Solution

to be completely honest I'm not sure. My first thoughts were to use something of the form

E = span{(1,0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}

Though I know that's wrong because that would be the span of any vector in R

^{4}

Sorry that my attempt isn't all that great and I'm probably way off, I'm just fairly confused with what I'm supposed to find as the span.

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