Suppose we have the vector space $V$ and the non-empty subspace $W$. I know there is a theorem that states that if $\bar{v}_1$ and $\bar{v}_2$ are vectors in a subspace $W$ then the vector $(\bar{v}_1 + \bar{v}_2)$ will also be in the subspace $W$. However is the converse true? Would having the vector $(\bar{v}_1 + \bar{v}_2)$ in $W$ imply that $\bar{v}_1$ and $\bar{v}_2$ are also in $W$?
Does having the vector $(\bar{v}_1 + \bar{v}_2)$ in the subspace $W$ imply that $\bar{v}_1$ and $\bar{v}_2$ are also in $W$?
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linear-algebra
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0No, this is not true. Consider the Nulspace W. If two vectors are in the Nulspace, then their sum is also in the Nulspace. But if a vector is in the Nulspace, then any arbitrary sum of if, is likely not to be in the Nulspace – 2017-01-18
2 Answers
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This is not true. Consider the trivial subspace that consist of only the zero vector.
Pick any non-zero vector, $v$, it is not inside $W$. but $v-v=0$
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Of course not. Let $P_W(\vec{v})$ be the projection of $\vec{v}$ onto subspace $W$.
Then as long as $$\vec{v}_1- P_W(\vec{v}_1)=- (\vec{v}_2-P_W(\vec{v}_2))$$ their sum will be in subspace $W$.