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.


Learn more about this topic:

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Vectors: Definition, Types & Examples

from Common Entrance Test (CET): Study Guide & Syllabus

Chapter 57 / Lesson 3
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