All Flashcards
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 .
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.
Explain how unit vectors are used to construct transformation matrices.
The transformed images of the standard basis vectors (unit vectors) become the columns of the transformation matrix.
What does the determinant of a transformation matrix tell you?
The absolute value of the determinant indicates the scaling factor of the area under the transformation. The sign indicates orientation (reflection).
Explain the concept of composing two linear transformations.
It involves applying one transformation after another. The matrix of the composition is the product of the individual transformation matrices.
What does it mean for a transformation to have an inverse?
It means there exists another transformation that 'undoes' the original transformation, mapping the output back to the original input.
Why is the order of matrix multiplication important in compositions?
Matrix multiplication is not commutative; therefore, the order in which transformations are applied matters. AB is not necessarily equal to BA.
Explain the geometric meaning of a negative determinant.
A negative determinant indicates that the transformation includes a reflection (flip) in addition to any possible dilation.
How do you find the matrix that represents a reflection over the y-axis?
The matrix is because it maps (1,0) to (-1,0) and (0,1) to (0,1).
How does the determinant relate to the area of a shape after a linear transformation?
The absolute value of the determinant gives the factor by which the area of any shape is scaled after the transformation.
Explain why not all matrices have an inverse.
A matrix has an inverse if and only if its determinant is non-zero. A zero determinant indicates that the transformation collapses space, making it impossible to 'undo'.
Describe the relationship between a matrix and its inverse.
The inverse of a matrix, when multiplied by the original matrix, results in the identity matrix. This represents 'undoing' the transformation.
What are the differences between rotation and reflection transformations?
Rotation: Preserves orientation, changes angle | Reflection: Reverses orientation (flips), maintains distance from axis.
What are the differences between dilation with a positive determinant and a negative determinant?
Positive Determinant: Dilation (expansion), no reflection | Negative Determinant: Dilation and reflection.
What are the differences between a transformation and its inverse?
Transformation: Maps input to output | Inverse: Maps output back to the original input, 'undoing' the transformation.
What are the differences between composing f(g(x)) and g(f(x))?
f(g(x)): Apply g first, then f | g(f(x)): Apply f first, then g. Order matters because matrix multiplication is not commutative.
What are the differences between a linear transformation and a non-linear transformation?
Linear Transformation: Preserves lines, origin fixed | Non-Linear Transformation: May curve lines, origin may shift.
What are the differences between scaling and shearing transformations?
Scaling: Changes the size of an object | Shearing: Distorts the shape of an object, shifting points parallel to an axis.
What are the differences between a matrix with a determinant of 1 and a matrix with a determinant of -1?
Determinant of 1: Preserves area and orientation | Determinant of -1: Preserves area but reverses orientation (reflection).
What are the differences between the identity matrix and the zero matrix?
Identity Matrix: Acts as '1' for matrix multiplication, leaves matrix unchanged | Zero Matrix: Acts as '0' for matrix addition, results in zero matrix upon multiplication.
What are the differences between row operations and column operations on a matrix?
Row Operations: Used to solve systems of equations, affect rows | Column Operations: Affect columns, related to linear combinations of column vectors.
What are the differences between orthogonal and non-orthogonal matrices?
Orthogonal Matrices: Columns (and rows) are orthonormal, preserve lengths and angles | Non-Orthogonal Matrices: Do not necessarily preserve lengths and angles.