Question

How to determine if a vector lies in a span of other vectors?

Ex:
a = <1,3,4>
b = <4,9,9>
c=<4,1,0>

Is c in the span of a and b?

Answer 1

`\mathbf c` lies in
`\mathrm{span}\{\mathbf a,\mathbf b\}` if
`\mathbf c` can be written as a linear combination of
`\mathbf a` and
`\mathbf b`.

In other words,
`\mathbf c\in\mathrm{span}\{\mathbf a,\mathbf b\}` if there exists scalars
`k_1,k_2\in\mathbb R` such that

`\mathbf c=k_1\mathbf a+k_2\mathbf b`

Vectors in
`\mathbb R^n` are equal if their components are equal:

`\langle4,1,0\rangle=k_1\langle1,3,4\rangle+k_2\langle4,9,9\rangle`

`\langle4,1,0\rangle=\langle k_1+4k_2,3k_1+9k_2,4k_1+9k_2\rangle`

`\implies\begin{cases}k_1+4k_2=4\\3k_1+9k_2=1\\4k_1+9k_2=0\end{cases}`

This system has no solutions, so
`\mathbf c` does not belong to
`\mathrm{span}\{\mathbf a,\mathbf b\}`.