Let u, v and w be three vectors in R^3. Then, | vedclass

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(D) For any vector $z \in R^3$ to be expressed as a linear combination $z = au + bv + cw$,the set of vectors ${u, v, w}$ must span the entire vector s

Vedclass · Accepted Answer

The vectors $u, v$ and $w$ are linearly independent

Vedclass · Answer

(D) For any vector $z \in R^3$ to be expressed as a linear combination $z = au + bv + cw$,the set of vectors ${u, v, w}$ must span the entire vector space $R^3$. Since $R^3$ is a $3$-dimensional space,any set of $3$ vectors that spans $R^3$ must be linearly independent. If the vectors are linearly dependent,they would lie in a plane or on a line,and thus could not represent every vector in $R^3$. Therefore,the condition is that $u, v$ and $w$ must be linearly independent. Since this option was not provided in the original list,option $D$ is the correct choice.

Let $u, v$ and $w$ be three vectors in $R^3$. Then,any vector $z \in R^3$ can be written as $z = au + bv + cw$ for some scalars $a, b$ and $c$ if and only if:

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