Does Multiplying Scalars and Vectors Always Yield Zero?

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SUMMARY

The discussion centers on the properties of scalar multiplication in vector spaces, specifically addressing the zero vector and its implications. It is established that for any scalar \( b \), multiplying it by the zero vector results in the zero vector, denoted as \( b \cdot \langle 0, 0, 0 \rangle = \langle b \cdot 0, b \cdot 0, b \cdot 0 \rangle = \langle 0, 0, 0 \rangle \). Furthermore, it is concluded that if \( b \cdot x = 0 \), then either \( b = 0 \) or \( x = 0 \) holds true in any vector space, not limited to \( \mathbb{R}^3 \) or \( \mathbb{R}^n \).

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kidia
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This one:
a)Let V be a vector space and let x be a vector in V.
i)Show that b0=0 for each scalar b.
ii)Show that if bx=0, then either b=0 or x=0
 
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Im assuming you mean the magnitude.
The zero vector has components <0,0,0>, the magnitude of this vector is [tex]\sqrt{0^2+0^2+0^2} = 0.[/tex]

b<x,y,z> = <bx,by,bz>

b<0,0,0> = <b0,b0,b0> = 0

ii) follows from i)
 
No, there is no reason to assume that kidia "means the magnitude" (and you don't then use "magnitude"). More importantly,there is no reason to assume that kidia meant R3 or any Rn. These theorems are true in any vector space.

kidia, what happens if you multiply a(x+ 0)? What does that tell you about a0?

Similarly, what happens if you multiply (a+ 0)x?
 

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