Final answer:
To determine the standard matrix for a vertical shear transformation that leaves a particular vector unchanged, we must consider how the basis vectors are transformed. The first basis vector does not change, while the second vector is sheared by adding a multiple of the first. The resulting standard matrix includes the transformed basis vectors as its columns.
Step-by-step explanation:
The question pertains to the determination of the standard matrix for a linear transformation, specifically a vertical shear transformation that maps into leaving a particular vector unchanged. To achieve this, we consider the impact of the transformation on the standard basis vectors in ℝ2. The first basis vector is (1,0), which typically represents the horizontal axis and would remain unchanged. The second basis vector is (0,1), which corresponds to the vertical axis and may be sheared.
In the context of a vertical shear, the first basis vector remains the same, while the second vector is transformed by adding a multiple of the first basis vector to it. This multiple is the shear factor. Let's denote the shear factor as 'k.' After the shear, the second basis vector becomes (k,1), representing the vertical component 'V' as a function of the horizontal component 't,' following the equation V = kt + vo, analogous to the expression for a straight line y = mx + b, where 'm' is the slope.
Considering this transformation, the standard matrix A for the transformation 't' is composed of the transformed basis vectors as its columns, which results in the matrix:
A = \[
\begin{bmatrix}
1 & k \\
0 & 1
\end{bmatrix}
\]
where 'k' is the vertical shear factor. This matrix A can be applied to any vector in ℝ2 to achieve the vertical shear transformation described.