(a) Determine whether ( 1 -2 -3 ) is in the span of ( 1 1 0 ) an

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(a) Determine whether ( 1 -2 -3 ) is in the...

(a) Determine whether $\left(\begin{array}{r}1 \\ -2 \\ -3\end{array}\right)$ is in the span of $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)$ and $\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right) . \quad$ (b) Is $\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right)$ in the span of $\left(\begin{array}{l}1 \\ 2 \\ 2\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 3 \\ 4\end{array}\right)$ ? (c) Is $\left(\begin{array}{r}3 \\ 0 \\ -1 \\ -2\end{array}\right)$ in the span of $\left(\begin{array}{l}1 \\ 2 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 3 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ 0 \\ 1 \\ -1\end{array}\right)$ ?

(a) Determine whether $\left(\begin{array}{r}1 \\ -2 \\ -3\end{array}\right)$ is in the span of $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)$ and $\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right) . \quad$ (b) Is $\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right)$ in the span of $\left(\begin{array}{l}1 \\ 2 \\ 2\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 3 \\ 4\end{array}\right)$ ?
(c) Is $\left(\begin{array}{r}3 \\ 0 \\ -1 \\ -2\end{array}\right)$ in the span of $\left(\begin{array}{l}1 \\ 2 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 3 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ 0 \\ 1 \\ -1\end{array}\right)$ ?

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Key Concepts

Consistency of Linear Systems

The consistency of a system of linear equations is crucial in verifying whether a vector is in the span of other vectors. A consistent system—one that has at least one solution—demonstrates that there are suitable scalar coefficients to express the vector as a combination of the given set.

Systems of Linear Equations

Determining if a vector is in the span of other vectors often reduces to solving a system of linear equations. Each equation represents a component of the vector, and the existence of a solution indicates that the vector can be written as a linear combination of the spanning set.

Linear Combination

A linear combination involves multiplying vectors by scalars and then adding the results together. This concept is used to determine whether a specific vector lies within the span of a set of vectors by checking if it can be expressed in terms of them.

Span

Span refers to the set of all possible linear combinations of a given set of vectors. It is a fundamental concept in linear algebra that describes how vectors can generate a subspace within a vector space. Understanding the span helps determine whether a particular vector can be represented as a combination of other vectors.

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