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In geometry, a vertical shear transformation is a linear transformation that shifts points vertically by a certain factor, while keeping the horizontal position (x-coordinate) unchanged. To understand this concept, imagine you have grid paper and you push the points vertically upwards or downwards but without altering their x-positions. This type of transformation is often used in computer graphics and animation.

When you perform a vertical shear by a factor of 2, each point on your Cartesian plane will have its y-coordinate altered as follows:

• The x-coordinate remains the same.
• The y-coordinate becomes: \( y' = y + 2x \)

This can be captured in matrix form using the transformation matrix:\[A_1 = \begin{pmatrix} 1 & 0 \ 2 & 1 \end{pmatrix}\]Here, any vector \((x, y)\) is transformed to \((x, y')\) using matrix multiplication, reinforcing the idea that only the y-coordinate is directly affected by the shear.

A horizontal shear transformation is similar to a vertical shear but affects the x-coordinate instead. This transformation alters the x-coordinate of points across the Cartesian plane, shifting them horizontally and maintaining the y-coordinate constant. Visualize this as you apply a push on a horizontal axis, causing points to glide sideways.

• The y-coordinate stays unchanged.
• The x-coordinate transforms as \( x' = x + 2y \)

The matrix representation for a horizontal shear by a factor of 2 is:\[A_2 = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix}\]In this matrix, the non-diagonal element '2' plays a crucial role in determining the amount of shear applied. When you multiply this matrix by a vector \((x, y)\), the result is a new vector \((x', y)\) where the x-coordinate reflects the shear, and the y-coordinate is left untouched.

Matrix multiplication is a critical operation when dealing with sequences of transformations. This is because multiple transformations can be condensed into a single matrix through multiplication, allowing all transformations to be applied simultaneously. In this exercise, we have a vertical shear followed by a horizontal shear.

The transformation matrices are represented as \(A_1\) for the vertical shear and \(A_2\) for the horizontal shear. To find a single matrix \(A\) that performs both transformations in order, you multiply these two matrices:\[A = A_2 \times A_1 = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \ 2 & 1 \end{pmatrix}\]The resulting matrix \(A\) is calculated by multiplying corresponding rows and columns. Each element of the new matrix is derived by summing the products of corresponding elements:

• The element in the first row, first column: \( (1 \cdot 1) + (2 \cdot 2) = 5 \)
• In the first row, second column: \( (1 \cdot 0) + (2 \cdot 1) = 2 \)
• For the second row, first column: \( (0 \cdot 1) + (1 \cdot 2) = 2 \)
• Finally, the second row, second column: \( (0 \cdot 0) + (1 \cdot 1) = 1 \)

Thus, the final matrix that accomplishes both transformations is:\[A = \begin{pmatrix} 5 & 2 \ 2 & 1 \end{pmatrix}\]By understanding matrix multiplication, you realize that combining basic transformations is often more efficient and straightforward, making it easier to represent complex transformations as a single matrix.