please show your work (1 pt) Which of the following sets of vectors are linearly independent? A....

Question:

please show your work (1 pt) Which of the following sets of vectors are linearly independent?

A. {(-1,8), (9, -1) }

B. {(6, -3)}

C. {(4, -7, 1, 7), 1, -5, 2, -9)}

D.{ (-2, -5, -8), (-9, 2, 4), (-7, 12, 7), (-2, 11 --7)}

E. {(0, 0)}

F. {(-7, 2, 00), -8, -5, -6), 1, -4, 9)}

G. {(-2, -4), (0, 0) }

H. {(-8, -9), 2, 4), (5, 7)}

Linearly Independent Vectors:

A set consisting of single non-zero vector is linearly independent.

Aset consisting of two vectors is said to be linarly independent if they are not multiples of each other.

On forming a matrix in the form of determinant, if value of determinant is non-zero then set of vectors is said to be linearly independent.

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent.

A set containing zero vector is linearly dependent.

Answer and Explanation:

A.

{eq}\left \{ \left ( -1,8 \right ),\left ( 9,-1 \right ) \right \} {/eq}

Linearly independent as they are not multiples of each other.

B.

{eq}\left \{ \left ( 6,-3 \right ) \right \} {/eq}

This set contains a single nonzero vector, so, linearly independent.

C.

{eq}\left \{ \left ( 4,-7,1,7 \right ),\left ( 1,-5,2,-9 \right ) \right \} {/eq}

Linearly independent as they are not multiples of each other.

D.

{eq}\left \{ \left ( -2,-5,-8 \right ),\left ( -9,2,4 \right ),\left ( -7,12,7 \right ),\left ( -2,11,-7 \right ) \right \} {/eq}

This set has four vectors such that each vector contains three entries.

This set is linearly dependent as it contains more vectors than entries in each vector.

E.

{eq}\left \{ \left ( 0,0 \right ) \right \} {/eq}

As this set contains single non-zero vector, so linearly dependent.

F.

{eq}\left \{ \left ( -7,2,0 \right ),\left ( -8,-5,-6 \right ),\left ( 1,-4,9 \right ) \right \} {/eq}

{eq}\left | \begin{matrix}-7&2&0\\-8&-5&-6\\1&-4& 9\end{matrix} \right |=-7(-45-24)-2(32+5)=-557\neq 0 {/eq}

So, this set of vectors is linearly independent as {eq}\left | \begin{matrix}-7&2&0\\-8&-5&-6\\1&-4& 9\end{matrix} \right |\neq 0 {/eq}

G.

{eq}\left \{ (0,0) \right \} {/eq}

This set contains single zero vector, so linearly dependent.

H.

{eq}{(-8, -9), (2, 4), (5, 7)} {/eq}

This set has three vectors such that each vector contains two entries.

This set is linearly dependent as it contains more vectors than entries in each vector.