Determine whether the given vectors from linearly dependent set or linearly independent set.
Question:
1) Show that u= \(\begin{bmatrix} 1 \\0 \\6 \end{bmatrix}\) can be expressed as a linear combination of
u1= \(\begin{bmatrix} 4 \\0 \\2 \end{bmatrix}\) u2=\(\begin{bmatrix} 0 \\5 \\ 0 \end{bmatrix}\) u3=\(\begin{bmatrix} 2 \\0 \\6 \end{bmatrix}\)
2) Determine whether the vectors given from a linearly dependent set or linearly independent set
a) u1= \(\begin{bmatrix} 2 \\2 \\0 \end{bmatrix}\) u2= \(\begin{bmatrix} 2 \\ -2 \\ 0 \end{bmatrix}\) u3= \(\begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}\)
b) u1= \(\begin{bmatrix} 3 \\-1 \\-1 \\2 \end{bmatrix}\) u2= \(\begin{bmatrix} 1 \\0 \\2 \\1 \end{bmatrix}\) u3= \(\begin{bmatrix} 3 \\-1 \\0 \\1 \end{bmatrix}\)
Summary:
a=\(\begin{bmatrix} 4 \\0 \\2 \end{bmatrix}\) + b= \(\begin{bmatrix} 0 \\5 \\ 0 \end{bmatrix}\) + c=
\(\begin{bmatrix} 4a \\0 \\2a \end{bmatrix}\) + b= \(\begin{bmatrix} 0 \\5b \\ 0 \end{bmatrix}\) + c=
\(\begin{bmatrix} 4a &0 &2a \\ 0& 5b& 0 \\ 2c &0 &6c \end{bmatrix}\) = \(\begin{bmatrix} 1 \\0 \\6 \end{bmatrix}\)
u1= \(\begin{bmatrix} 3 \\-1 \\-1 \\2 \end{bmatrix}\) u2= \(\begin{bmatrix} 1 \\0 \\2 \\1 \end{bmatrix}\) u3= \(\begin{bmatrix} 3 \\-1 \\0 \\1 \end{bmatrix}\)
