Linear Transformations Checkpoint 1 Review

Front

What is a linear transformation?

Back

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Front

What is the image of a point under a linear transformation?

Back

The image of a point under a linear transformation is the result of applying the transformation to that point, resulting in a new point in the transformed space.

Front

What is a preimage in the context of linear transformations?

Back

The preimage is the original point in the input space that is transformed to a given image point in the output space.

Front

How do you find the image of a point (x, y) under a transformation T(x, y) = (ax + by, cx + dy)?

Back

Substitute the values of x and y into the transformation equation to calculate the new coordinates.

Front

What is the determinant of a transformation matrix?

Back

The determinant is a scalar value that provides information about the scaling factor of the transformation and whether it preserves orientation.

Front

How does the determinant affect the area of a shape under a linear transformation?

Back

The area of the image of a shape is equal to the area of the preimage multiplied by the absolute value of the determinant of the transformation matrix.

Front

What is the transformation matrix for T(x, y) = (2x + 3y, 5x - y)?

Back

The transformation matrix is \( \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix} \).

Front

If the image of a point is known, how can you find the preimage?

Back

To find the preimage, you need to apply the inverse of the transformation matrix to the image point.

Front

What is the effect of a transformation matrix with a determinant of zero?

Back

A transformation matrix with a determinant of zero collapses the space into a lower dimension, meaning it cannot be inverted.

Front

How do you determine if a transformation is linear?

Back

A transformation is linear if it satisfies two properties: T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalar c.