Which of the below is/are not true with respect to the specified mappings in R2? Here, 12 = [e₁ e₂] is a 2 x 2 identity matrix. A horizontal shear leaves the first entry of each vector unchanged. B. Reflection across the line x₂ = x, interchanges the entries of a vector. C Projection onto x₁-axis makes the second entry of the vector zero, not changing the first entry. Reflection through the origin negates both entries of a vector. the vector e axis maps onto e1 Reflection across the x₂- and the vector e₂ onto (-e₂). Under a vertical shear, the image of the vector e, is the vector e₁ itself.

- This statement is **true**. In a horizontal shear transformation, only the second entry of each vector is modified, while the first entry remains unchanged.

B. Reflection across the line x₂ = x interchanges the entries of a vector.

- This statement is **false**. Reflection across the line x₂ = x does not interchange the entries of a vector. It reflects the vector across the line, but the entries remain the same.

C. Projection onto the x₁-axis makes the second entry of the vector zero, not changing the first entry.

- This statement is **true**. When a vector is projected onto the x₁-axis, its second entry is set to zero, while the first entry remains unchanged.

D. Reflection through the origin negates both entries of a vector.

- This statement is **true**. Reflection through the origin reflects the vector across the origin, resulting in both entries being negated.

E. The vector e axis maps onto e₁.

- This statement is **false**. The vector e₁ maps onto itself, not the entire e-axis.

F. Reflection across the x₂-axis maps the vector e₂ onto (-e₂).

- This statement is **true**. Reflection across the x₂-axis flips the sign of the second entry of a vector, so the vector e₂ will be mapped onto (-e₂).

G. Under a vertical shear, the image of the vector e is the vector e₁ itself.

- This statement is **false**. Under a vertical shear transformation, the image of the vector e is not e₁ itself. The vertical shear will modify both entries of the vector e.

To summarize:

- Statement A is true.

- Statement B is false.

- Statement C is true.

- Statement D is true.

- Statement E is false.

- Statement F is true.

- Statement G is false.

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