In each part, determine whether multiplication by A is a one-to-one linear transformation. (a) (b) (c)

Data Analysis Formulas Midterm 1 ▯ ▯ ▯ (▯▯▯▯) • Sample Variance- = ▯▯▯ (▯▯▯) ▯ (▯▯▯▯)▯ • Sample Standard Deviation- = ▯▯▯ (▯▯▯) c • P(E) (Independent event)= 1-P(E ) ▯(▯∩▯) • Conditional Probability= (/) = ▯(▯) • Probability= + ( ) = 1 ▯ • Probability= (/)() + (/ )( ) ▯ ▯ • Independence= ( ∩ )= P(A)*P(B) • Independence= ∪ =P(A)+P(B)- ( ∩ ) • Mutually Exclusive= ( ∩ )= 0 • Mutually Exclusive= ∪ =P(A)+P(B) • Chebyshev’s Inequality: Inner bound is greater than 1 − ▯ ▯ ▯ ▯ • One sided: outer bound is less than ▯ ▯ ▯▯ • Normal Data Set: • Empirical Rules: • 1. If data is normally distributed, then 68% of data points fall within 1 standard deviation • 2. If data is normally distributed, then 95% of data points fall within 2 standard deviati