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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 :

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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