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What is the formula for a general linear transformation?
$T(x) = Ax$, where A is the transformation matrix and x is the input vector.
What is the formula for preserving vector addition in a linear transformation?
$T(u + v) = T(u) + T(v)$
What is the formula for preserving scalar multiplication in a linear transformation?
$T(cu) = cT(u)$
Given matrix $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and vector $v = \begin{bmatrix} x \\ y \end{bmatrix}$, what is $Av$?
$Av = \begin{bmatrix} a_{11}x + a_{12}y \\ a_{21}x + a_{22}y \end{bmatrix}$
If $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and $X = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}$, what is the formula for $AX$?
$AX = \begin{bmatrix} a_{11}v_1 + a_{12}v_2 & a_{11}v_2 + a_{12}v_3 & \cdots & a_{11}v_n + a_{12}v_n \\ a_{21}v_1 + a_{22}v_2 & a_{21}v_2 + a_{22}v_3 & \cdots & a_{21}v_n + a_{22}v_n \end{bmatrix}$
What is the general form of a reflection matrix across the x-axis?
$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
What is the general form of a reflection matrix across the y-axis?
$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$
If T is a linear transformation and 0 is the zero vector, what is T(0)?
$T(0) = 0$
How do you find the transformation matrix A, given T([1,0]) = [a,c] and T([0,1]) = [b,d]?
$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$
How do you find the combined transformation matrix of two transformations, T1 and T2, applied sequentially (T2 after T1)?
If T1 is represented by matrix A and T2 by matrix B, the combined transformation is BA.
How to find the transformation matrix given the transformation of basis vectors?
1. Determine the images of the standard basis vectors (e.g., [1, 0] and [0, 1]). 2. Place these images as columns in the transformation matrix.
How to determine the image of a vector after a sequence of transformations?
1. Find the transformation matrix for each transformation. 2. Multiply the matrices in the reverse order of the transformations. 3. Multiply the resulting matrix by the vector.
How to determine if a transformation is linear given its definition?
1. Check if T(u + v) = T(u) + T(v) for arbitrary vectors u and v. 2. Check if T(cu) = cT(u) for an arbitrary vector u and scalar c. 3. If both conditions are met, the transformation is linear.
How to find the matrix that represents a reflection across the x-axis followed by a given transformation T?
1. Find the matrix A representing the transformation T. 2. Find the matrix R representing reflection across the x-axis: $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$. 3. Multiply the matrices: AR.
How to find the area of a transformed shape after applying a linear transformation?
1. Find the transformation matrix A. 2. Calculate the determinant of A, |A|. 3. Multiply the original area by |A| to get the transformed area.
How to find the image of a vector v under the linear transformation T(x) = Ax?
Perform the matrix multiplication Av. The resulting vector is the image of v under T.
How to find the transformation matrix A if you know how T transforms two linearly independent vectors?
Let the two vectors be v1 and v2, and their images be T(v1) and T(v2). If v1 and v2 are the standard basis vectors, the images directly form the columns of A. Otherwise, solve for A such that Av1 = T(v1) and Av2 = T(v2).
How to find a matrix that represents a transformation that first scales a vector by a factor of k, then applies a transformation T?
1. Find the matrix A representing the transformation T. 2. Find the scaling matrix S = $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$. 3. Multiply the matrices: AS.
How to find a matrix that represents a transformation that rotates a vector by 90 degrees counterclockwise, then applies a transformation T?
1. Find the matrix A representing the transformation T. 2. Find the rotation matrix R = $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. 3. Multiply the matrices: AR.
How do you determine if a transformation is a linear transformation?
Check if it satisfies the two properties of linear transformations: T(u+v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalar c.
Define a linear transformation.
A function that preserves vector addition and scalar multiplication: T(u + v) = T(u) + T(v) and T(cu) = cT(u).
What is a transformation matrix?
A matrix that represents a linear transformation, such that T(x) = Ax.
What is the zero vector?
A vector where all components are zero; also called the null vector.
What does 'preserves addition' mean in the context of linear transformations?
T(u + v) = T(u) + T(v): The transformation of the sum of two vectors is the sum of their individual transformations.
What does 'preserves scalar multiplication' mean in the context of linear transformations?
T(cu) = cT(u): The transformation of a scalar times a vector is the scalar times the transformation of the vector.
What is the form of a single vector in R²?
A 2x1 matrix (column vector) represented as $\begin{bmatrix} x \\ y \end{bmatrix}$.
What is the form of a set of n vectors in R²?
A 2xn matrix represented as $\begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}$.
What is the result of applying a linear transformation to the zero vector?
The result is always the zero vector.
What is the significance of a 2x2 matrix A in a linear transformation from R² to R²?
The matrix A is the transformation matrix that uniquely defines the linear transformation L, such that L(v) = Av.
Define the term 'image' in the context of linear transformations.
The resulting vector after applying a linear transformation to an input vector.