91Ó°ÊÓ

Additivity is one of the key properties that a linear transformation must satisfy. To understand additivity, let's consider it in the context of our transformation \( \phi: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \). We want to check if this transformation is linear. For additivity, this means if we take two vectors \( v_1 \) and \( v_2 \) and apply the transformation to their sum, the result should be the same as transforming each vector individually and then adding the results.

Mathematically, this means:

• \( \phi(v_1 + v_2) = \phi(v_1) + \phi(v_2) \)

When checking this, our transformation gives us \( \phi(v_1 + v_2) = (v_1 + v_2) + u \). On the other hand, calculating \( \phi(v_1) + \phi(v_2) \), we get \( v_1 + u + v_2 + u = v_1 + v_2 + 2u \). For these to be equal, the vector \( u \) must be zero, as that would make the additional \( u \) terms disappear.

• The conclusion is that additivity holds only when \( u = 0 \).

Homogeneity is another crucial property for linear transformations. It ensures that if you multiply a vector by a scalar before transforming it, the result is the same as if you transformed the vector first and then multiplied by the scalar. This concept is pivotal in understanding linearity because it assures that distributions scale appropriately.

Let's write out what homogeneity means for our transformation \( \phi \):

• \( \phi(cv) = c\phi(v) \) for any scalar \( c \).

For our particular map, applying a scalar \( c \) to a vector \( v \), we have \( \phi(cv) = cv + u \) but \( c\phi(v) = c(v + u) = cv + cu \). These two expressions will only be equal if \( u = cu \), which implies \( u = 0 \) unless \( c \) is specifically chosen to counteract any value of \( u \) not equal to zero.

Since step 1 found \( u = 0 \) for additivity, this condition is satisfied for homogeneity. The property of homogeneity is thus met when \( u \) remains zero, reinforcing the linearity.

Vector spaces form the backbone of linear algebra. In our problem, we are dealing with transformations within the vector space \( \mathbb{R}^2 \). Understanding vector spaces is fundamental to comprehending linear transformations because they provide the structure in which vectors reside.

A vector space is defined by certain properties, such as:

• Closure under addition and scalar multiplication
• Existence of a zero vector
• Associativity and commutativity of vector addition

Within this space, you can perform operations like adding vectors or scaling them by numbers (scalars). The linear transformation we analyzed, \( \phi \), is a function that maps vectors within this vector space while striving to maintain structural integrity in terms of operations like addition and scalar multiplication.

When we stated the necessity for \( u = (0,0) \) for linearity, we were referring to the effect of transformations that preserve the vector space's intrinsic properties. Thus, vector spaces are essential for ensuring that transformations do not distort the space's fundamental nature, which is precisely what our exercise aimed to establish.