Consider the matrices A through E below. A=[ 0.6 0.8 0.8 -0.6

step 1 All right, so first we have matrix B represents the scaling, then matrix C represents an orthogo

Consider the matrices A through E below. A=[ ...

Consider the matrices $A$ through $E$ below. \[\begin{aligned}A=\left[\begin{array}{rr}0.6 & 0.8 \\0.8 & -0.6\end{array}\right], \quad B=\left[\begin{array}{ll}3 & 0 \\0 & 3\end{array}\right] \\C=\left[\begin{array}{rr}0.36 & -0.48 \\ -0.48 & 0.64\end{array}\right], \quad D=\left[\begin{array}{rr}-0.8 & 0.6 \\-0.6 & -0.8\end{array}\right] \\ E=\left[\begin{array}{rr}1 & 0 \\-1 & 1\end{array}\right]\end{aligned}\] Fill in the blanks in the sentences below. We are told that there is a solution in each case. Matrix ___ represents a scaling. Matrix ___ represents an orthogonal projection. Matrix ___ represents a shear. Matrix ___ represents a reflection. Matrix ___ represents a rotation.

Consider the matrices $A$ through $E$ below.
\[\begin{aligned}A=\left[\begin{array}{rr}0.6 & 0.8 \\0.8 & -0.6\end{array}\right], \quad B=\left[\begin{array}{ll}3 & 0 \\0 & 3\end{array}\right] \\C=\left[\begin{array}{rr}0.36 & -0.48 \\
-0.48 & 0.64\end{array}\right], \quad D=\left[\begin{array}{rr}-0.8 & 0.6 \\-0.6 & -0.8\end{array}\right] \\
E=\left[\begin{array}{rr}1 & 0 \\-1 & 1\end{array}\right]\end{aligned}\]
Fill in the blanks in the sentences below. We are told that there is a solution in each case.
Matrix ___ represents a scaling.
Matrix ___ represents an orthogonal projection.
Matrix ___ represents a shear.
Matrix ___ represents a reflection.
Matrix ___ represents a rotation.

Key Concepts

Rotation

Rotation is a linear transformation that turns every vector about the origin by a given angle while preserving lengths and angles. A rotation matrix is orthogonal and has a determinant equal to 1, meaning it is an isometry that maintains the overall geometric structure, simply rotating the vectors to a new position.

Reflection

Reflection is a transformation that 'flips' vectors over a certain line (or plane in higher dimensions) by reversing the sign of components perpendicular to that line while keeping parallel components unchanged. The reflection matrix is orthogonal with a determinant of -1, ensuring that distances are preserved while the orientation is reversed.

Shear

A shear transformation shifts points in a fixed direction, with the amount of shift proportional to another coordinate. The matrix representing a shear typically has ones along the main diagonal and non-zero off-diagonal entries, which 'shear' the shape without changing its area, altering angles while preserving parallelism.

Orthogonal Projection

An orthogonal projection is a linear transformation that maps vectors onto a subspace, such as a line or plane, in such a way that the difference between the original vector and its image is orthogonal to the subspace. The projection is characterized by a matrix that is both symmetric and idempotent (i.e., applying the projection twice gives the same result as applying it once).

Scaling

Scaling is a linear transformation that enlarges or reduces objects uniformly by multiplying all components of a vector by a constant. In matrix terms, a scaling matrix is typically diagonal, where identical entries down the main diagonal (with zeros elsewhere) represent the uniform scaling factor across all dimensions.

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