Calculating Linear Span: Vector a1 (-7, 8, 5) and Line Equation

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Homework Help Overview

The problem involves demonstrating that the linear span of the vector a1 = (-7, 8, 5) corresponds to a specific line equation in three-dimensional space.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the definition of the linear span and its implications for a single vector. There is an exploration of how to express points in the span using scalar multiplication.

Discussion Status

Some participants have provided definitions and clarifications regarding the linear span, while others are attempting to formulate the relationship between the vector and the line equation. The discussion is ongoing with various interpretations being explored.

Contextual Notes

There is a noted uncertainty about how to initiate the problem, and participants are encouraged to define terms and explore relationships without providing direct solutions.

dracolnyte
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Homework Statement


(i)Show that the linear span of the vector a1 = (-7, 8, 5) is the line whose equation is
x/(-7) = y/8 = z/5

The Attempt at a Solution


The problem is, I don't know where or how to start.
 
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Define the 'linear span of a vector'. Look it up if you have to.
 
The linear span of a vector is the collection of all linear combinations of the vectors a1, a2..., ak
 
Ok, so for one vector v1 it's the set of all vectors k*v1 for k any real number, right?
 
Ya, and then?
 
You are supposed to be helping here. Guess. (x,y,z) is a point in the span of (-7, 8, 5). What's it equal to?
 
Okay so
-7k = x
8k = y
5k = z
 
Got it, thank you so much. I'll be back with more for sure lol.
 

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