MHB Tyler's question at Yahoo Answers (linear independence)

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If the set {w1, ... wm} is linearly independent in a vector space V, then the corresponding set of coordinate vectors {[w1]B, ... [wm]B} is also linearly independent in Rn. This is demonstrated using properties of vector representation in a fixed basis B, including linearity and the uniqueness of representation for the zero vector. By assuming a linear combination of the coordinate vectors equals zero, it leads to a conclusion that the original vectors must also combine to zero, contradicting their independence. Therefore, the linear independence of the original set guarantees the linear independence of the transformed set in Rn. This establishes a fundamental relationship between linear independence in different vector spaces.
Fernando Revilla
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Here is the question:

Show that if {w1, ... wm} is linearly independent in V, then {[w1]B , ... [wm]B} is linearly independent in Rn.

Here is a link to the question:

Linear algebra help please please? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Tyler,

Fixing a basis $B=\{w_1,\ldots,w_m\}$ of a real vector space $V$ we know that

$(i)\;[x+y]_B=[x]_B+[y]_B$ or all $x,y\in V$.
$(ii)\;[\lambda x]_B=\lambda [x]_B$ for all $\lambda \in \mathbb{R}$.
$(iii)\;[x]_B=(0,\ldots ,0)\Leftrightarrow x=0$.

Suppose $\lambda_1 [w_1]_B +\ldots +\lambda_m [w_m]_B=(0,\ldots ,0)$. Using $(i)$ and $(ii)$:
$$[\lambda_1 w_1 +\ldots +\lambda_m w_m]_B=(0,\ldots, 0)$$
Using $(iii)$: $\lambda_1 w_1 +\ldots +\lambda_m w_m=0$.

By hypothesis $ \{w_1,\ldots ,w_m\}$ are linearly independent, so $\lambda_1=\ldots =\lambda_m=0$. This proves that $\{[w_1]_B,\ldots,[w_m]_B\}$ are linearly independent.
 
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