Please work this problem out. I want to make sure I worked it out correctly. Suppose that V is a vector space with basis B = {b¡ | i E I} and S is asubspace of V. Let {B1,., Bk} be a partition of B. Then is it true that%3DkS = O(sn (B;))ミ=1What if Sn (B) # {0} for all i?

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Description: Please work this problem out. I want to make sure I worked it out correctly. Suppose that V is a vector space with basis B = {b¡ | i E I} and S is asubspace of V. Let {B1,., Bk} be a partition of B. Then is it true that%3DkS = O(sn (B;))ミ=1What if S

Given that V is a vector space with basis B=bi|i∈I and S is a subspace of V. Let B1,B2,B3,…,Bk be a…

Answered: Suppose that V is a vector space with…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 24CM

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Please work this problem out. I want to make sure I worked it out correctly.

Suppose that V is a vector space with basis B = {b¡ | i E I} and S is a
subspace of V. Let {B1,., Bk} be a partition of B. Then is it true that
%3D
k
S = O(sn (B;))
ミ=1
What if Sn (B) # {0} for all i?

Expert Solution

Step 1

Given that $V$ is a vector space with basis $B = \{b_{i} | i \in I\}$ and $S$ is a subspace of $V$ .

Let $\{B_{1}, B_{2}, B_{3}, \dots, B_{k}\}$ be a partition of $B$ .

Claim: $S \neq \oplus_{i = 1}^{k} (S \cap <B_{i}>)$

Consider $V = ℝ^{2}$ and $S = (<1, 1>)$ with $B_{1} = \{e_{1}\}$ and $B_{2} = \{e_{2}\}$ .

Then note that $S \cap B_{i} = \{0\}$ for $i = 1, 2$ .

Therefore, $(S \cap B_{1}) + (S \cap B_{2}) = \{0\}$ but $S \neq \{0\}$ .

Hence, it follows that $S \neq \oplus_{i = 1}^{k} (S \cap <B_{i}>)$ .

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