Select all of the linear transformations from $\mathbb{R}^3$ to $\mathbb{R}^3$ that are invertible. There may be more than one correct answer.

A. Identity transformation

B. Projection onto the $xz$-plane

C. Reflection in the $y$-axis

D. Rotation about the $x$-axis

E. Dilation by a factor of 6

F. Projection onto the $z$-axis

Question

27-02-2024
Mathematics

College

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Select all of the linear transformations from $\mathbb{R}^3$ to $\mathbb{R}^3$ that are invertible. There may be more than one correct answer.

A. Identity transformation

B. Projection onto the $xz$-plane

C. Reflection in the $y$-axis

D. Rotation about the $x$-axis

E. Dilation by a factor of 6

F. Projection onto the $z$-axis

Answer :

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tiwariakdi tiwariakdi · Answer 1 · 2024-01-16T05:27:20-05:00

A linear transformation is a function from one vector space to another that preserves each vector space's underlying (linear) structure. A linear transformation can also be referred to as a linear operator or map.

The correct answers are A, D, and E.

A linear transformation from ℝ3 to ℝ3 is invertible if and only if it is bijective, meaning it is both one-to-one and onto.

The following linear transformations are invertible:

A. Identity transformation: The identity transformation maps every vector to itself and is one-to-one, so it is invertible.

D. Rotation about the x-axis: A rotation about the x-axis is a bijective linear transformation that maps each point in ℝ3 to another point in ℝ3 in a one-to-one manner, so it is invertible.

E. Dilation by a factor of 6: A dilation by a factor of 6 scales every vector by 6, and the scaling factor is nonzero, so this transformation is bijective and invertible.

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