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Example
Created on 2024-06-20Asked by Olivia Lewis (Solvelet student)
Determine if the following transformation is a linear transformation: T:R2→R2defined byT(x,y)=(2x−y,3x+4y) T: \mathbb{R}^2 \to \mathbb{R}^2 \quad \text{defined by} \quad T(x, y) = (2x - y, 3x + 4y)

Solution

To determine if the given transformation T T is a linear transformation, we need to check if it satisfies the two properties of linearity: 1. Additivity: T(u+v)=T(u)+T(v) T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) 2. Homogeneity: T(cu)=cT(u) T(c\mathbf{u}) = cT(\mathbf{u}) Given T(x,y)=(2x−y,3x+4y) T(x, y) = (2x - y, 3x + 4y) . Let u=(x1,y1) \mathbf{u} = (x_1, y_1) and v=(x2,y2) \mathbf{v} = (x_2, y_2) . Check additivity: T(u+v)=T((x1,y1)+(x2,y2))=T(x1+x2,y1+y2) T(\mathbf{u} + \mathbf{v}) = T((x_1, y_1) + (x_2, y_2)) = T(x_1 + x_2, y_1 + y_2) =(2(x1+x2)−(y1+y2),3(x1+x2)+4(y1+y2)) = (2(x_1 + x_2) - (y_1 + y_2), 3(x_1 + x_2) + 4(y_1 + y_2)) =(2x1+2x2−y1−y2,3x1+3x2+4y1+4y2) = (2x_1 + 2x_2 - y_1 - y_2, 3x_1 + 3x_2 + 4y_1 + 4y_2) Now compute T(u)+T(v) T(\mathbf{u}) + T(\mathbf{v}) : T(u)=T(x1,y1)=(2x1−y1,3x1+4y1) T(\mathbf{u}) = T(x_1, y_1) = (2x_1 - y_1, 3x_1 + 4y_1) T(v)=T(x2,y2)=(2x2−y2,3x2+4y2) T(\mathbf{v}) = T(x_2, y_2) = (2x_2 - y_2, 3x_2 + 4y_2) T(u)+T(v)=(2x1−y1,3x1+4y1)+(2x2−y2,3x2+4y2) T(\mathbf{u}) + T(\mathbf{v}) = (2x_1 - y_1, 3x_1 + 4y_1) + (2x_2 - y_2, 3x_2 + 4y_2) =(2x1−y1+2x2−y2,3x1+4y1+3x2+4y2) = (2x_1 - y_1 + 2x_2 - y_2, 3x_1 + 4y_1 + 3x_2 + 4y_2) =(2x1+2x2−y1−y2,3x1+3x2+4y1+4y2) = (2x_1 + 2x_2 - y_1 - y_2, 3x_1 + 3x_2 + 4y_1 + 4y_2) Since T(u+v)=T(u)+T(v) T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) , additivity is satisfied. Check homogeneity: T(cu)=T(c(x1,y1))=T(cx1,cy1) T(c\mathbf{u}) = T(c(x_1, y_1)) = T(cx_1, cy_1) =(2(cx1)−(cy1),3(cx1)+4(cy1)) = (2(cx_1) - (cy_1), 3(cx_1) + 4(cy_1)) =(c(2x1−y1),c(3x1+4y1)) = (c(2x_1 - y_1), c(3x_1 + 4y_1)) =c(2x1−y1,3x1+4y1) = c(2x_1 - y_1, 3x_1 + 4y_1) =cT(x1,y1) = cT(x_1, y_1) Since T(cu)=cT(u) T(c\mathbf{u}) = cT(\mathbf{u}) , homogeneity is satisfied. Therefore, T(x,y)=(2x−y,3x+4y) T(x, y) = (2x - y, 3x + 4y) is a linear transformation. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Jack Brown on Solvelet
1. Determine whether the linear transformation T:R2→R2 T: \mathbb{R}^2 \to \mathbb{R}^2 defined by T(x,y)=(2x+y,3y) T(x, y) = (2x + y, 3y) is onto.2. Find the matrix representation of the linear transformation T:R3→R3T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 that reflects points across the plane x=yx = y.
DefinitionA transformation function is a function of two vector spaces that follows the model of vector spaces. For a linear transformation T, the vectors u (and v and the scalar c) have T(u + v) = T(u) + T(v) and T(cu) = cT(u). For example, the transformation T is a linear function defined by the matrix (x, y) = (2x, 3y).
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