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GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 6

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Consider a set of $n$ linearly independent vectors $\left\{\vec{w}_1, \ldots, \vec{w}_n\right\} \in \mathbb{R}^n$. A vector $\vec{u} \in \mathbb{R}^n$ will:

  • Option $1.$ Always be a linear combination of $\left\{\vec{w}_1, \ldots, \vec{w}_n\right\}$
  • Option $2.$ Never be a linear combination of $\left\{\vec{w}_1, \ldots, \vec{w}_n\right\}$

Enter the correct option as the numeric number. That is if option $2$ is correct then enter $\text{“2”}.$

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In a vector space $R^{n}$, any set of n linearly independent vectors forms the basis vectors.

i.e A set of n linearly independent vectors belonging to the vector space $R^{n}$ can span the entire

space of $R^{n}$.

So a vector $\overrightarrow{u} $ $\epsilon$ $R^{n}$ can always be expressed as the linear combination

of n linearly independent vectors {$\overrightarrow{w_{1}} $,…,$\overrightarrow{w_{n}} $} $\epsilon$ $R^{n}$.
 

Option1 is correct

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{w1,w2,w3,…,wn}∈Rn and the set is linearly independent and the set consists of exactly n vectors so this set is the minimum and enough to fill the space of Rn.If you include one vector u∈Rn ,then u always can be written as the linear combination of these n vectors .so option 1 is correct.
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