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.

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.

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.

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.

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.

Perform the matrix multiplication Av. The resulting vector is the image of v under T.

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).

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.

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.

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.

$T(x) = Ax$, where A is the transformation matrix and x is the input vector.

$T(u + v) = T(u) + T(v)$

$T(cu) = cT(u)$

$Av = \begin{bmatrix} a_{11}x + a_{12}y \\ a_{21}x + a_{22}y \end{bmatrix}$

$AX = \begin{bmatrix} a_{11}v_1 + a_{12}v_2 & a_{11}v_2 + a_{12}v_3 & \cdots & a_{11}v_n + a_{12}v_n \\ a_{21}v_1 + a_{22}v_2 & a_{21}v_2 + a_{22}v_3 & \cdots & a_{21}v_n + a_{22}v_n \end{bmatrix}$

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

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

$T(0) = 0$

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

If T1 is represented by matrix A and T2 by matrix B, the combined transformation is BA.

A matrix 'A' acts on a vector 'x' through matrix multiplication (Ax) to produce a transformed vector. The matrix encodes the scaling, rotation, and shearing inherent in the transformation.

Matrix multiplication allows us to apply a linear transformation to a vector or a set of vectors efficiently. It also allows us to compose multiple transformations by multiplying their matrices.

A linear transformation always maps the zero vector to the zero vector. This property can be used to quickly check if a transformation is linear.

Linear transformations can stretch, rotate, shear, or reflect geometric shapes, but they preserve lines and parallelism. Squares can become parallelograms, for example.

Verify that it preserves vector addition and scalar multiplication: T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalar c.

The columns of the transformation matrix are the images of the standard basis vectors (e.g., [1, 0] and [0, 1] in R²) after the transformation.

Multiply the transformation matrix by the vector. The resulting vector is the image of the original vector under the transformation.

It mirrors a vector or shape across a line (e.g., the x-axis or y-axis). The distance to the line of reflection remains the same, but the direction is reversed.

Applying one linear transformation after another. The combined transformation is represented by the matrix product of the individual transformation matrices (in reverse order of application).

Solving a system of linear equations can be interpreted as finding the pre-image of a vector under a linear transformation. The transformation matrix represents the coefficients of the system.