Question

In Exercises 5–8, determine if the columns of the matrix form a

linearly independent set. Justify each answer.

5. \(\left[ {\begin{array}{*{20}{c}}0&{ - 8}&5\\3&{ - 7}&4\\{ - 1}&5&{ - 4}\\1&{ - 3}&2\end{array}} \right]\)

Short answer

The columns are linearly independent.

Step-by-step solution

Write the condition for the linear independence of the columns of the matrix

The vectors are said to be linearly independent if the equation \(A{\bf{x}} = 0\) has a trivial solution, where A is the matrix and xis the vector.

Write the matrix in the augmented form

Consider the matrix \(\left[ {\begin{array}{*{20}{c}}0&{ - 8}&5\\3&{ - 7}&4\\{ - 1}&5&{ - 4}\\1&{ - 3}&2\end{array}} \right]\). As there are three columns in the matrix, there should be three entries in the vector.

Thus, the matrix equation is \(\left[ {\begin{array}{*{20}{c}}0&{ - 8}&5\\3&{ - 7}&4\\{ - 1}&5&{ - 4}\\1&{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\\0\\0\end{array}} \right]\), and it is in \(A{\bf{x}} = {\bf{0}}\) form.

Write the augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&{\bf{0}}\end{array}} \right]\) as shown below:

\(\left[ {\begin{array}{*{20}{c}}0&{ - 8}&5&0\\3&{ - 7}&4&0\\{ - 1}&5&{ - 4}&0\\1&{ - 3}&2&0\end{array}} \right]\)

Convert the augmented matrix in the echelon form

In the echelon form, at the top of the left-most column, the leading entry should be non-zero.

Interchange rows one and four.

\(\left[ {\begin{array}{*{20}{c}}0&{ - 8}&5&0\\3&{ - 7}&4&0\\{ - 1}&5&{ - 4}&0\\1&{ - 3}&2&0\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&{ - 3}&2&0\\3&{ - 7}&4&0\\{ - 1}&5&{ - 4}&0\\0&{ - 8}&5&0\end{array}} \right]\)

Add \( - 3\) times row one to row two to eliminate the \(3{x_1}\) term from the second equation. Add rows one and three to eliminate the \({x_1}\) term from the third equation.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&2&0\\3&{ - 7}&4&0\\{ - 1}&5&{ - 4}&0\\0&{ - 8}&5&0\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&{ - 3}&2&0\\0&2&{ - 2}&0\\0&2&{ - 2}&0\\0&{ - 8}&5&0\end{array}} \right]\)

Add \( - 1\) time row two to row three to eliminate the \( - {x_1}\) term from the third equation. Add \( - 4\) times row two to row four to eliminate the \( - 8{x_2}\) term from the fourth equation. Interchange rows three and four.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&2&0\\0&2&{ - 2}&0\\0&0&0&0\\0&0&{ - 3}&0\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&{ - 3}&2&0\\0&2&{ - 2}&0\\0&0&{ - 3}&0\\0&0&0&0\end{array}} \right]\)

Multiply row two by \(\frac{1}{2}\) and row three by \( - \frac{1}{3}\).

\(\left[ {\begin{array}{*{20}{c}}1&0&2&0\\0&1&0&0\\0&0&1&0\\0&0&0&0\end{array}} \right]\)

Add \( - 2\) times row three to row one to eliminate the \(2{x_3}\) term from the first equation.

\(\left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0\end{array}} \right]\)

Mark the pivot positions in the matrix

Mark the non-zero leading entries in columns 1, 2 and 3.

Check for the linear independence of the matrix

According to the pivot positions in the obtained matrix, there are no free variables.

Thus, the homogeneous equation has a trivial solution (only one solution of the equation), which means the vectors are linearly independent.