Answer :
The linear transformations B (identity), D (rotation about z-axis), and E (dilation by 5) are the ones that are invertible from [tex]\( \mathbb{R}^3 \) to \( \mathbb{R}^3 \)[/tex].
Linear transformations from [tex]\( \mathbb{R}^3 \) to \( \mathbb{R}^3 \)[/tex] that are invertible are the identity transformation (B), rotation about the z-axis (D), and dilation by a factor of 5 (E).
In linear algebra, a transformation from [tex]\( \mathbb{R}^3 \) to \( \mathbb{R}^3 \)[/tex] is invertible if and only if its determinant is non-zero. Let's analyze each option:
A. Trivial transformation [tex]\( T(\vec{v}) = \vec{0} \)[/tex]:
This transformation maps every vector to the zero vector. Its determinant is 0, hence it is not invertible.
B. Identity transformation [tex]\( T(\vec{v}) = \vec{v} \)[/tex]:
This transformation leaves every vector unchanged. Its determinant is 1, so it is invertible.
C. Reflection in the origin:
This transformation flips vectors through the origin. Its determinant is -1, so it is not invertible.
D. Rotation about the z-axis:
Rotation transformations preserve lengths and angles, and their determinant is 1. Thus, rotations about any axis are invertible.
E. Dilation by a factor of 5:
This transformation scales every vector by a factor of 5. Its determinant is [tex]\( 5^3 = 125 \)[/tex], which is non-zero. Hence, dilation by a factor of 5 is invertible.
F. Projection onto the xy -plane:
This transformation projects vectors onto the xy-plane, collapsing the z -component. Its determinant is 0, so it is not invertible.
Therefore, the linear transformations B (identity), D (rotation about z-axis), and E (dilation by 5) are the ones that are invertible from [tex]\( \mathbb{R}^3 \) to \( \mathbb{R}^3 \)[/tex].
Complete Question:
Select all of the linear transformations from [tex]$\mathbb{R}^3$[/tex] to [tex]$\mathbb{R}^3$[/tex] that are invertible. There may be more than one correct answer.
A. Trivial transformation (i.e. [tex]$T(\vec{v})=\overrightarrow{0}$[/tex] for all [tex]$\vec{v}$[/tex] )
B. Identity transformation (i.e. [tex]$T(\vec{v})=\vec{v}$[/tex] for all [tex]$\vec{v}$[/tex] )
C. Reflection in the origin
D. Rotation about the z-axis
E. Dilation by a factor of 5
F. Projection onto the x y-plane
Final answer:
To determine which linear transformations from ℝ3 to ℝ3 are invertible, we need to check their determinants. The identity transformation, reflection in the y-axis, rotation about the x-axis, and dilation by a factor of 6 are all invertible.
Explanation:
A linear transformation from ℝ3 to ℝ3 is invertible if and only if its determinant is non-zero. Let's analyze each option:
A. Identity transformation: The identity transformation is defined as T(v⃗) = v⃗ for all v⃗. Its determinant is 1, so it is invertible.
B. Projection onto the xz-plane: The projection onto the xz-plane has determinant 0, so it is not invertible.
C. Reflection in the y-axis: The reflection in the y-axis has determinant -1, so it is invertible.
D. Rotation about the x-axis: The rotation about the x-axis has determinant 1, so it is invertible.
E. Dilation by a factor of 6: The dilation by a non-zero factor has determinant 6^3 = 216, so it is invertible.
F. Projection onto the z-axis: The projection onto the z-axis has determinant 0, so it is not invertible.
