Linear Transformations Checkpoint 1 Review

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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.

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What is the standard form of a linear transformation in two dimensions?

Back

The standard form of a linear transformation in two dimensions can be represented as T(x, y) = (ax + by, cx + dy), where a, b, c, and d are constants.

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How do you find the image of a point under a linear transformation?

Back

To find the image of a point (x, y) under a linear transformation T, substitute the coordinates into the transformation's formula.

Front

What is the matrix representation of a linear transformation?

Back

The matrix representation of a linear transformation T(x, y) can be expressed as a 2x2 matrix that transforms the vector (x, y) when multiplied.

Front

What is the inverse of a linear transformation?

Back

The inverse of a linear transformation T is another transformation T^{-1} such that T^{-1}(T(x, y)) = (x, y) for all points (x, y) in the domain.

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How do you calculate the inverse of a transformation given its matrix?

Back

To calculate the inverse of a transformation given its matrix, you can use the formula for the inverse of a 2x2 matrix: if A = [[a, b], [c, d]], then A^{-1} = (1/(ad-bc)) * [[d, -b], [-c, a]].

Front

What is the identity matrix?

Back

The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere, denoted as I, and it acts as the multiplicative identity in matrix multiplication.

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How do you verify if one transformation is the inverse of another?

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To verify if one transformation is the inverse of another, multiply their matrix representations and check if the result is the identity matrix.

Front

What is the effect of a linear transformation on the area of geometric shapes?

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A linear transformation can change the area of geometric shapes by a factor equal to the absolute value of the determinant of the transformation's matrix.

Front

What is the determinant of a 2x2 matrix?

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The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc.