(a) Show that the vectors ( 1 0 2 1 ),( -2 3 -1 1 ),

step 1 Show that these vectors are linearly independent in first -degree polynomial with real coefficie

(a) Show that the vectors ( 1 0 2 ...

(a) Show that the vectors $\left(\begin{array}{l}1 \\ 0 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}-2 \\ 3 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -2 \\ 1 \\ -1\end{array}\right)$ are linearly independent. (b) Whi $)$ of the following vectors are in their span? (i) $\left(\begin{array}{l}1 \\ 1 \\ 2 \\ 1\end{array}\right)$, (ii) $\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 0\end{array}\right)$, (iii) $\left(\begin{array}{l}0 \\ 1 \\ 0 \\ 0\end{array}\right)$, (iv) $\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right)$. (c) Suppose $\mathbf{b}=(a, b, c, d)^T$ lies in their span. What conditions must $a, b, c, d$ satisfy?

(a) Show that the vectors $\left(\begin{array}{l}1 \\ 0 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}-2 \\ 3 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -2 \\ 1 \\ -1\end{array}\right)$ are linearly independent. (b) Whi $)$ of the following vectors are in their span? (i) $\left(\begin{array}{l}1 \\ 1 \\ 2 \\ 1\end{array}\right)$, (ii) $\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 0\end{array}\right)$, (iii) $\left(\begin{array}{l}0 \\ 1 \\ 0 \\ 0\end{array}\right)$, (iv) $\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right)$.
(c) Suppose $\mathbf{b}=(a, b, c, d)^T$ lies in their span. What conditions must $a, b, c, d$ satisfy?

Key Concepts

Parameterization and Necessary Conditions

When expressing a vector as a linear combination of other vectors, the components of the vector must satisfy certain equations, often leading to conditions or constraints among them. These relations, derived from equating the corresponding components of the linear combination, are essential for identifying which vectors belong to the span and for characterizing the subspace spanned by a set of vectors.

Systems of Linear Equations

To determine span membership or to verify linear independence, one typically sets up a system of linear equations based on the components of the vectors. Solving these equations reveals whether the coefficients in the linear combination can produce a given vector or show that only the trivial solution exists in the case of linear independence.

Linear Independence

This concept refers to a set of vectors where no vector in the set can be written as a combination of the others. In the context of a vector space, the only solution to a linear equation formed by setting a linear combination of these vectors equal to the zero vector is that all coefficients must be zero. This property is key to determining the minimal spanning set or basis for the space.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. It forms a subspace of the overall vector space, and identifying whether another vector lies in the span involves determining if that vector can be obtained by a suitable combination of the given vectors.

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