Question

Which of the following are linear transformations? Converting from meters to kilometers Squaring each side to find the area Converting from ounces to pounds Taking the square root of each person's height. Multiplying all numbers by 2 and then adding 5 Converting temperature from Fahrenheit to Centigrade

Short answer

Linear transformations: converting meters to kilometers, converting ounces to pounds.

Step-by-step solution

Understanding Linear Transformations

A function or a transformation is linear if it satisfies two main properties: additivity and homogeneity. This means for a transformation \( T \), we have \( T(u + v) = T(u) + T(v) \) and \( T(cu) = cT(u) \) for any vectors \( u, v \) and scalar \( c \).

Analyze Conversion from Meters to Kilometers

The conversion from meters to kilometers is linear. It satisfies the properties: \( f(x) = \frac{x}{1000} \). Thus, \( f(u+v) = \frac{u+v}{1000} = \frac{u}{1000} + \frac{v}{1000} = f(u) + f(v) \), and \( f(cu) = \frac{cu}{1000} = c\frac{u}{1000} = cf(u) \).

Analyze Squaring Each Side to Find the Area

Squaring each side is not a linear transformation. The transformation \(f(x) = x^2\) does not satisfy additivity as \( (u+v)^2 eq u^2 + v^2 \), hence it is non-linear.

Analyze Conversion from Ounces to Pounds

The conversion from ounces to pounds is linear. This is a simple scaling transformation: \( f(x) = \frac{x}{16} \) or \( f(x) = 0.0625x \). It satisfies both additivity and homogeneity.

Analyze Taking the Square Root of Each Person's Height

Taking the square root of heights is non-linear. The transformation \(f(x) = \sqrt{x}\) does not satisfy additivity: \( \sqrt{u+v} eq \sqrt{u} + \sqrt{v} \).

Analyze Multiplying All Numbers by 2 and Then Adding 5

This transformation \(f(x) = 2x + 5\) is non-linear due to the constant (+5). It does not satisfy the additivity property \( f(u+v) = 2(u+v) + 5 eq 2u + 5 + 2v + 5 = f(u) + f(v) \).

Analyze Converting Temperature from Fahrenheit to Centigrade

This transformation \( f(x) = \frac{5}{9}(x-32) \) is linear. We can rearrange it to fit \( f(x) = \frac{5}{9}x - \frac{160}{9} \). While it contains a constant, the conversion itself is based on a linear relationship between inputs and outputs.

Key concepts

Additivity Property

A key aspect of linear transformations is the additivity property. This means that for any transformation function \( T \), when you apply it to the sum of two inputs \( u \) and \( v \), it should equal the sum of the transformations of each input separately.
This can be described mathematically as \( T(u + v) = T(u) + T(v) \). It's like saying that combining two groups of numbers and then transforming them is the same as transforming each group individually and then combining the results.

• The conversion from meters to kilometers is a perfect example of this. If you have two distances, say 500 meters and 300 meters, adding them gives you 800 meters. Whether you convert 800 meters to kilometers in one go or convert 500 and 300 meters separately and then sum the results, the outcome remains the same.


• In contrast, squaring a number or taking the square root does not satisfy this property, which highlights their nature as non-linear transformations.

Homogeneity Property

Homogeneity is another important aspect of linear transformations. It indicates that if you scale an input by a factor \( c \), the transformed output should also be scaled by the same factor. Mathematically, this is expressed as \( T(cu) = cT(u) \).
This is akin to stretching a shape; no matter how big or small you make it, the proportional relationship remains the same.

• For instance, converting ounces to pounds is homogeneous. If you have \( x \) ounces, converting it is as simple as multiplying by a constant factor. Scaling the ounces will equally scale the pounds.


• However, transformations like \( f(x) = 2x + 5 \) do not maintain this property because of the constant addition of 5.

Scaling Transformations

Scaling transformations are all about changing the size without altering the original properties. They are a subset of linear transformations where a function scales inputs consistently.
For example, converting meters to kilometers involves scaling by the factor of \( \frac{1}{1000} \). This scaling means if you double the number of meters, you will double the number of kilometers too.

• This ensures both the additivity and homogeneity properties are met, hence confirming a linear relationship.


• This is different from transformations like \( 2x + 5 \) which include an additional step beyond simple scaling, thus becoming non-linear transformations.

Non-Linear Transformations

Non-linear transformations don't adhere to the rules of additivity and homogeneity. They involve processes where the output isn't directly proportional to the input.
These transformations can produce curves, varying results in different intervals, or distortions in the data.

• For instance, squaring a number transforms it in a non-linear fashion, changing its proportion relative to its input. This is mostly evident as the input grows larger.


• Taking a square root is also non-linear, providing smaller growth increments with larger inputs, quite the opposite of squaring.

Temperature Conversion

Temperature conversion between Fahrenheit and Centigrade is fascinating because it showcases a type of linear transformation. Despite its formula \( f(x) = \frac{5}{9}(x-32) \) having a constant term, the transformation remains linear relative to its inputs and outputs.
This means it holds a direct, proportional relationship without distortion even if the equation is adjusted with constants.

• This proves that constants, when part of properly structured equations, don't always create non-linear effects. The conversion maintains the straight-line relationship characteristic of linear transformations.