Let T : ℝ^3 →ℝ^3 be the transformation that projects each vector 𝐱=(x1, x2, x3) onto the plane x

step 1 So we're given a transformation, and we want to show the T's linear. So in order to do that, we

Let T : ℝ^3 →ℝ^3 be the transformation that projects each...

Let $T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be the transformation that projects each vector $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$ onto the plane $x_{2}=0,$ so $T(\mathbf{x})=$ $\left(x_{1}, 0, x_{3}\right) .$ Show that $T$ is a linear transformation.

Key Concepts

Linear Transformation

A linear transformation is a mapping between vector spaces that preserves the operations of vector addition and scalar multiplication. This means for any vectors u and v and any scalar c, the transformation satisfies T(u + v) = T(u) + T(v) and T(c · u) = c · T(u), ensuring that the structure of the vector space is maintained under the transformation.

Additivity (Preservation of Vector Addition)

The additivity property is one of the essential criteria for a linear transformation. It states that the image of the sum of two vectors is equal to the sum of the images of the individual vectors. This property ensures that combining vectors in the domain yields a corresponding combination in the codomain.

Homogeneity (Preservation of Scalar Multiplication)

Homogeneity, also known as scalar multiplication preservation, is another key property for establishing linearity. It requires that multiplying a vector by a scalar before applying the transformation has the same effect as applying the transformation first and then multiplying by the scalar, i.e., T(c · v) = c · T(v) for any scalar c. This property ensures that scaling is consistent under the transformation.

Projection onto a Subspace

Projection is a specific type of linear transformation where vectors from a higher-dimensional space are 'projected' onto a subspace. In this context, the transformation takes any vector and maps it onto a subspace that satisfies a particular condition (such as lying on a plane). Projection operations often simplify problems by reducing the dimensions while preserving the linear structure.

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