Question

If \(A\) is \(3 \times 3,\) show that the image of the line in \(\mathbb{R}^{3}\) through \(\mathbf{p}_{0}\) with direction vector \(\mathbf{d}\) is the line through \(A \mathbf{p}_{0}\) with direction vector \(A\) d, assuming that \(A \mathbf{d} \neq \mathbf{0} .\) What happens if \(A \mathbf{d}=\mathbf{0} ?\)

Short answer

The transformed line is through \( A\mathbf{p}_0 \) with direction \( A\mathbf{d} \); if \( A\mathbf{d} = \mathbf{0} \), it collapses to \( A\mathbf{p}_0 \).

Step-by-step solution

Define the Line in \( \mathbb{R}^3 \)

A line in \( \mathbb{R}^3 \) through a point \( \mathbf{p}_0 \) with direction vector \( \mathbf{d} \) can be described by the parametric equation: \( \mathbf{r}(t) = \mathbf{p}_0 + t \mathbf{d} \), where \( t \) is a scalar.

Transform the Line using Matrix \( A \)

To find the image of the line under the transformation by matrix \( A \), we apply \( A \) to each point on the line. This gives the new line: \( A\mathbf{r}(t) = A(\mathbf{p}_0 + t \mathbf{d}) = A\mathbf{p}_0 + t(A\mathbf{d}) \).

Analyze the Transformed Line

The equation \( A\mathbf{p}_0 + t(A\mathbf{d}) \) describes a line through \( A\mathbf{p}_0 \) with a direction vector \( A\mathbf{d} \). This shows that the direction vector of the transformed line is the image of the original direction vector under \( A \).

Consider the Case \( A\mathbf{d} = \mathbf{0} \)

If \( A\mathbf{d} = \mathbf{0} \), then the transformed equation becomes \( \mathbf{r}'(t) = A\mathbf{p}_0 + t\mathbf{0} = A\mathbf{p}_0 \). This suggests that every point on the line maps to the single point \( A\mathbf{p}_0 \), indicating the line collapses to a point.

Key concepts

Image of a Line

In geometry, the concept of the image of a line under a transformation is crucial to understanding how lines behave when subjected to certain changes, like those involving matrices. When we talk about the **image of a line**, we're referring to the new line that results from applying a transformation to every point on the original line. If you picture drawing a line on a piece of paper and then moving it elsewhere without changing its shape, that moved line represents the image of the original line. For transformations described by a matrix, each point on the line goes through a shift and a change according to how the matrix alters the space around it.

In our specific exercise, this involves a 3x3 matrix that shifts both the starting point of a line in 3D space and its direction, creating a new line that's the transformed version of the original. If the transformation is correct, the image should accurately reflect this change. Distorting a line into a point is an unusual scenario but shows how drastic transformations can become when specific conditions are met.

Direction Vectors

Direction vectors are fundamental in describing lines in geometry. They give us a way to express which direction a line moves through space. If you think of a direction vector as an arrow pointing from one point to another, it helps us understand what path the line will take. For a line defined with a direction vector \( \mathbf{d} \) starting from point \( \mathbf{p}_0 \), we can extend this line infinitely in the direction the vector is pointing.

In terms of matrix transformations, when a line undergoes transformation, its direction vector also transforms. The direction vector's transformation depends on how the matrix alters the vector's components in space. In our exercise, if the matrix outputs a non-zero vector \( A\mathbf{d} \), it means we have a valid direction for the new line. However, if the matrix transforms the direction vector \( \mathbf{d} \) into a zero vector, the notion of direction disappears, and the line collapses to a single point.

Matrix Transformations

Matrix transformations allow us to modify geometric figures within a given space. Imagine matrices as a set of instructions that tell you how to move and resize every point on a line or shape. By applying a matrix, we can deform figures, rotate them, move them from one place to another, or even squash them into different forms.

In the context of our exercise, the 3x3 matrix \( A \) transforms lines in 3D space by shifting their base points and altering their direction vectors. The new line formed after transformation not only moves to start from a new location but angles itself based on how its direction vector is altered. The line positions change according to how the matrix multiplies with the original points and direction vectors, crafting a new version of the familiar line aimed towards a different path.

Parametric Equations

Parametric equations are a powerful tool in geometry for describing lines and curves. They use one or more parameters, such as time \( t \), to express the coordinates of the points on a line or curve. The beauty of parametric equations is their ability to describe motion or a path in a very flexible and robust way.

In our studied problem, a line is described using a parametric equation \( \mathbf{r}(t) = \mathbf{p}_0 + t \mathbf{d} \), where \( \mathbf{p}_0 \) is the starting point and \( \mathbf{d} \) is the direction vector. As \( t \) varies, \( \mathbf{r}(t) \) traces out all the points on the line. When a transformation matrix \( A \) applies to this parametric equation, it shifts the starting point to \( A\mathbf{p}_0 \) and alters the direction of the line via the new direction vector \( A\mathbf{d} \). This redefined parametric equation portrays the image, displaying the changed path through the space.