Final answer:
To determine whether a transformation is linear from r2 into r3, we need to check if it satisfies the preservation of vector addition and scalar multiplication. This can be done by verifying the conditions for all vectors in r2 or by checking the transformation of a basis for r2. A specific example is provided to illustrate the process.
Step-by-step explanation:
A linear transformation is a function that maps vectors from one vector space to another, preserving the vector addition and scalar multiplication operations. To determine whether a transformation is linear, we need to check if it satisfies two conditions: the preservation of vector addition and scalar multiplication. If a transformation T satisfies T(u + v) = T(u) + T(v) and T(cu) = cT(u), where u and v are vectors and c is a scalar, then T is a linear transformation.
In this case, the transformation is from r2 (2-dimensional space) to r3 (3-dimensional space). To check if a given transformation is linear, we can either verify the two conditions for all vectors in r2, or we can check the transformation of a basis for r2. If the transformation of the basis vectors satisfies the two conditions, then the transformation is linear.
Let's consider an example. Suppose the transformation is defined by T(x, y) = (x + 2y, -3x + y, 4x - y). We can check if this transformation is linear by verifying the preservation of vector addition and scalar multiplication:
- Vector addition: T((x1, y1) + (x2, y2)) = T((x1 + x2, y1 + y2)) = ((x1 + x2) + 2(y1 + y2), -3(x1 + x2) + (y1 + y2), 4(x1 + x2) - (y1 + y2)).
- Scalar multiplication: T(c(x, y)) = T((cx, cy)) = ((cx) + 2(cy), -3(cx) + (cy), 4(cx) - (cy)).
If we simplify these expressions using basic algebra, we can see that both the vector addition and scalar multiplication properties are satisfied. Therefore, the given transformation is linear from r2 into r3.