Question

5. Determine, with proof, whether each of the following subsets S of a vector space V is linearly dependent or independent: a) V = R. S = {(2, 8.-1.4), (3.2. 4.0), (-1,-5, 2, 3), (0.0.7, 2)} 1112×2

Answer 1

a) Given $S=\{(2,8,-1,4),(3,2,4,0),(-1,-5,2,3),(0,0,7,2)\}$

Now clearly vectors (3,2,4,0) & (0,0,7,2) are Linearily Independent as one vector has Non zero component corresponds to same place Another vector has zero component

Now suppose

These four equation give different value of constant a&b which is not possible

is Linearily Independent

Next suppose

Solving equation (1)&(2) we obtain

a=1/10,c=-13/20 Substitute in equation (4) we get b=14/5

But values obtained in equation (1),(2)&(4) doesn't satisfies equation (3)

is Linearily Independent

b) $Let \ v_1=\begin{bmatrix} 1 &0 \\ 0 & -1 \end{bmatrix},v_2=\begin{bmatrix} 0 &1 \\ 1& 0 \end{bmatrix}\&v_3=\begin {bmatrix} 1&1\\-1&1\end {bmatrix}$

Clealy {As }

Now

If $av_1+bv_2=v_3 \\ a\begin {bmatrix} 1&0\\0&-1\end {bmatrix} +b\begin{bmatrix} 0&1\\1&0\end {bmatrix} =\begin {bmatrix} 1&1\\-1&1\end {bmatrix} \\==>\begin{bmatrix} a&b\\b&-a\end{bmatrix} =\begin {bmatrix} 1&1\\-1&1\end {bmatrix} \\ this \ give \ following \ equations \\ a=1,a=-1,b=1,b=-1$

Which make no sense

===>