Define Linear Transformation.

A function that moves and stretches vectors, keeping lines straight and the origin fixed.

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Define Linear Transformation.

A function that moves and stretches vectors, keeping lines straight and the origin fixed.

What is a Transformation Matrix?

A matrix that represents a linear transformation, dictating how vectors are transformed.

Define 'Image of a Vector'.

The transformed vector resulting from multiplying a vector by a transformation matrix.

What is a Rotation Matrix?

A matrix that rotates a vector by a specific angle, typically counterclockwise.

Define Determinant.

For a 2x2 matrix, it's the value calculated as $a_{11}a_{22} - a_{12}a_{21}$ .

What is Dilation Magnitude?

The absolute value of the determinant, indicating the scaling of areas under a transformation.

Define Composition of Transformations.

Applying one linear transformation after another, where the output of the first becomes the input of the second.

What is an Inverse Transformation?

A transformation that 'undoes' the effect of another transformation.

Define Inverse Matrix.

A matrix that, when multiplied by the original matrix, results in the identity matrix.

What is the Identity Matrix?

A matrix that, when multiplied by any matrix, leaves the matrix unchanged. Denoted as I.

How do you find the transformation matrix given the transformation of standard basis vectors?

Determine where <1, 0> and <0, 1> are mapped. 2. Place the transformed <1, 0> as the first column and the transformed <0, 1> as the second column of the matrix.

How do you rotate a vector by a given angle using a matrix?

Determine the angle of rotation. 2. Construct the rotation matrix. 3. Multiply the rotation matrix by the vector.

How do you find the area of a region after a linear transformation?

Find the determinant of the transformation matrix. 2. Take the absolute value of the determinant. 3. Multiply the original area by the absolute value of the determinant.

How do you determine the matrix for a composite transformation?

Find the matrices for each individual transformation. 2. Multiply the matrices in the correct order (right to left).

How do you determine if a matrix has an inverse?

Calculate the determinant of the matrix. 2. If the determinant is non-zero, the matrix has an inverse.

How do you apply a composite transformation to a vector?

Find the composite transformation matrix. 2. Multiply the composite matrix by the vector.

How do you find the matrix for a reflection over the x-axis followed by a rotation of 45 degrees?

Find the matrix for reflection over the x-axis. 2. Find the matrix for a 45-degree rotation. 3. Multiply the rotation matrix by the reflection matrix.

How do you find the inverse of a 2x2 matrix?

Calculate the determinant (ad-bc). 2. Swap 'a' and 'd', negate 'b' and 'c'. 3. Divide each element by the determinant.

How do you find the matrix that scales x by 2 and y by 3?

Create a diagonal matrix. 2. Put 2 in the top-left entry and 3 in the bottom-right entry. 3. The matrix is $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ .

How do you verify that two matrices are inverses of each other?

Multiply the two matrices together. 2. If the result is the identity matrix, then they are inverses.

What is the general form of a 2D transformation matrix?

$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

What is the formula for a counterclockwise rotation matrix by angle θ?

$\begin{bmatrix} cos(θ) & -sin(θ) \\ sin(θ) & cos(θ) \end{bmatrix}$

What are the transformation equations for a counterclockwise rotation by angle θ?

$x' = xcos(θ) - ysin(θ)$ , $y' = xsin(θ) + ycos(θ)$

What is the formula for a clockwise rotation matrix by angle θ?

$\begin{bmatrix} cos(-θ) & -sin(-θ) \\ sin(-θ) & cos(-θ) \end{bmatrix}$

What is the formula for the determinant of a 2x2 matrix?

For matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is $ad - bc$ .

If matrix A represents transformation f and matrix B represents transformation g, what matrix represents the composition g(f(x))?

If a linear transformation L is given by L(v) = Av, what is its inverse L⁻¹?

L⁻¹(v) = A⁻¹v

What is the matrix for reflection over the y-axis?

$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

What is the matrix for reflection over the x-axis?

$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

What is the formula for scaling by a factor of k in both x and y directions?

$\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$

All Flashcards

Define Linear Transformation.

A function that moves and stretches vectors, keeping lines straight and the origin fixed.

What is a Transformation Matrix?

A matrix that represents a linear transformation, dictating how vectors are transformed.

Define 'Image of a Vector'.

The transformed vector resulting from multiplying a vector by a transformation matrix.

What is a Rotation Matrix?

A matrix that rotates a vector by a specific angle, typically counterclockwise.

Define Determinant.

For a 2x2 matrix, it's the value calculated as $a_{11}a_{22} - a_{12}a_{21}$ .

What is Dilation Magnitude?

The absolute value of the determinant, indicating the scaling of areas under a transformation.

Define Composition of Transformations.

Applying one linear transformation after another, where the output of the first becomes the input of the second.

What is an Inverse Transformation?

A transformation that 'undoes' the effect of another transformation.

Define Inverse Matrix.

A matrix that, when multiplied by the original matrix, results in the identity matrix.

What is the Identity Matrix?

A matrix that, when multiplied by any matrix, leaves the matrix unchanged. Denoted as I.

How do you find the transformation matrix given the transformation of standard basis vectors?

Determine where <1, 0> and <0, 1> are mapped. 2. Place the transformed <1, 0> as the first column and the transformed <0, 1> as the second column of the matrix.

How do you rotate a vector by a given angle using a matrix?

Determine the angle of rotation. 2. Construct the rotation matrix. 3. Multiply the rotation matrix by the vector.

How do you find the area of a region after a linear transformation?

Find the determinant of the transformation matrix. 2. Take the absolute value of the determinant. 3. Multiply the original area by the absolute value of the determinant.

How do you determine the matrix for a composite transformation?

Find the matrices for each individual transformation. 2. Multiply the matrices in the correct order (right to left).

How do you determine if a matrix has an inverse?

Calculate the determinant of the matrix. 2. If the determinant is non-zero, the matrix has an inverse.

How do you apply a composite transformation to a vector?

Find the composite transformation matrix. 2. Multiply the composite matrix by the vector.

How do you find the matrix for a reflection over the x-axis followed by a rotation of 45 degrees?

Find the matrix for reflection over the x-axis. 2. Find the matrix for a 45-degree rotation. 3. Multiply the rotation matrix by the reflection matrix.

How do you find the inverse of a 2x2 matrix?

Calculate the determinant (ad-bc). 2. Swap 'a' and 'd', negate 'b' and 'c'. 3. Divide each element by the determinant.

How do you find the matrix that scales x by 2 and y by 3?

Create a diagonal matrix. 2. Put 2 in the top-left entry and 3 in the bottom-right entry. 3. The matrix is $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ .

How do you verify that two matrices are inverses of each other?

Multiply the two matrices together. 2. If the result is the identity matrix, then they are inverses.

What is the general form of a 2D transformation matrix?

$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

What is the formula for a counterclockwise rotation matrix by angle θ?

$\begin{bmatrix} cos(θ) & -sin(θ) \\ sin(θ) & cos(θ) \end{bmatrix}$

What are the transformation equations for a counterclockwise rotation by angle θ?

$x' = xcos(θ) - ysin(θ)$ , $y' = xsin(θ) + ycos(θ)$

What is the formula for a clockwise rotation matrix by angle θ?

$\begin{bmatrix} cos(-θ) & -sin(-θ) \\ sin(-θ) & cos(-θ) \end{bmatrix}$

What is the formula for the determinant of a 2x2 matrix?

For matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is $ad - bc$ .

If matrix A represents transformation f and matrix B represents transformation g, what matrix represents the composition g(f(x))?

If a linear transformation L is given by L(v) = Av, what is its inverse L⁻¹?

L⁻¹(v) = A⁻¹v

What is the matrix for reflection over the y-axis?

$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

What is the matrix for reflection over the x-axis?

$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

What is the formula for scaling by a factor of k in both x and y directions?

$\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$