Make a Change of Variable, X = Py, That Transforms $$Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }$$

question 20

Multiple Choice

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product
term. Give P and the new quadratic form.
- $$Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }$$

• A)
• \[\mathrm { P } = \left[ \begin{array} { r r }
• 1 & - 2 \\
• 2 & 1
• \end{array} \right] ; 9 \mathrm { y } _ { 1 } + 4 \mathrm { y } _ { 2 }\]
• B)
• \[\mathrm { P } = \left[ \begin{array} { c c }
• 1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \\
• 2 / \sqrt { 5 } & 1 / \sqrt { 5 }
• \end{array} \right] ; 9 \mathrm { y } _ { 1 } ^ { 2 } + 4 \mathrm { y } _ { 2 } ^ { 2 }\]
• C)
• \[P = \left[ \begin{array} { r r }
• 1 & - 2 \\
• 2 & 1
• \end{array} \right] ; 9 y _ { 1 } ^ { 2 } + 4 y _ { 2 } ^ { 2 }\]
• D)
• \[P = \left[ \begin{array} { r r }
• - 1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \\
• 2 / \sqrt { 5 } & 1 / \sqrt { 5 }
• \end{array} \right] ; - 9 y _ { 1 } ^ { 2 } - 4 y _ { 2 } ^ { 2 }\]

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Definitions:

Alternative Hypothesis

The hypothesis that proposes a significant difference, relationship, or effect, and is accepted if the null hypothesis is rejected.

Simple Linear Regression

A statistical method that models the relationship between a dependent variable and one independent variable, assuming a linear relationship.

68-95-99.7 Rule

A statistical rule stating that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Observed Responses

Observed responses refer to the actual outcomes or reactions recorded in an experiment or survey, following exposure to certain conditions or stimuli.