Find a Basis for the Null Space of the Matrix $$A = \left[ \begin{array} { r r r r r } 1 & 0 & - 4 & 0 & - 4 \\0 & 1 & 2 & 0 & 2 \\0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0\end{array} \right]$$

question 68

Multiple Choice

Find a basis for the null space of the matrix.
- $$A = \left[ \begin{array} { r r r r r } 1 & 0 & - 4 & 0 & - 4 \\0 & 1 & 2 & 0 & 2 \\0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0\end{array} \right]$$

• A)
• \( \left\{\left[\begin{array}{r}-4 \\ 2 \\ 1 \\ 0 \\ 0\end{array}\right],\left[\begin{array}{r}-4 \\ 2 \\ 0 \\ -1 \\ 1\end{array}\right]\right\} \)
• B)
• \[\left\{ \left[ \begin{array} { r }
• - 4 \\
• 2 \\
• 1 \\
• 0 \\
• 0
• \end{array} \right] , \left[ \begin{array} { r }
• - 4 \\
• 2 \\
• 0 \\
• - 1 \\
• 1
• \end{array} \right] \right\}\]
• C)
• \(\left\{\left[\begin{array}{r}
• 1 \\
• 0 \\
• -4 \\
• 0 \\
• -4
• \end{array}\right],\left[\begin{array}{l}
• 0 \\
• 1 \\
• 2 \\
• 0 \\
• 2
• \end{array}\right]\right\}\)
• D)
• \(\left\{\left[\begin{array}{l}
• 1 \\
• 0 \\
• 0 \\
• 0
• \end{array}\right],\left[\begin{array}{l}
• 0 \\
• 1 \\
• 0 \\
• 0
• \end{array}\right],\left[\begin{array}{l}
• 0 \\
• 0 \\
• 1 \\
• 0
• \end{array}\right]\right\}\)

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Definitions:

Mode

The value that appears most frequently in a data set.

Mean

The average of a set of numbers, calculated by summing the numbers and then dividing by the count of those numbers.

Median

The middle value in a data set when the values are arranged in ascending or descending order, effectively splitting the dataset into two halves.

Range

The difference between the highest and lowest values in a data set, showing the spread or dispersion of the data points.