#AP Pre-Calculus: Linear Transformations & Matrices - The Night Before 🚀

Hey! Let's get you prepped for the exam. We'll break down linear transformations and matrices, making sure you're confident and ready. Let's do this!

#4.12 Linear Transformations and Matrices

#What's a Linear Transformation? 🤔

Think of a linear transformation as a special kind of function that takes an input vector and spits out an output vector, but it does so in a way that respects vector addition and scalar multiplication. It's like a well-behaved function that keeps things nice and linear. ⚙️

• Preserves Addition: If you have two vectors, u and v, then T(u + v) = T(u) + T(v). It's like the transformation doesn't mix them up; it transforms each separately and then adds the results.
• Preserves Scalar Multiplication: If you have a vector u and a scalar c, then T(cu) = cT(u). The transformation doesn't change the scaling factor; it transforms the vector and then scales the result. 🎁

For any input vector x, the output vector T(x) is found by multiplying x by the transformation matrix A: T(x) = A * x. Each part of the output vector is just a sum of scaled parts of the input vector. That's why it's called linear – it keeps the linearity of the input vector. ↕️

Source: Medium

#♾️ Vectors: Zero, Single, and Many

• Zero Vector: Also called the null vector, it's a vector where all components are zero. It's like the origin in our vector space. 0️⃣

  • Why? Because T(0) = T(0*u) = 0*T(u) = 0. The zero vector always stays at the origin after a linear transformation.

• Single Vector in R²: Represented as a 2 x 1 matrix (a column vector). It looks like [x; y], where x and y are the coordinates of the vector in the x-y plane. This is how we usually write vectors in R². 📓

• Set of n Vectors in R²: Represented as a 2 x n matrix. It's like putting n column vectors side by side: [v1, v2, ..., vn]. This gives you a matrix with 2 rows and n columns. 📚

Source: Google Sites

#✔️ How Linear Transformations Work

For any linear transformation L from R² to R², there's a unique 2x2 matrix A such that L(v) = Av for all vectors v in R². This matrix A is the key to the transformation. 💈

• A is the transformation matrix of L. It holds all the info about how L stretches, rotates, or reflects vectors in R². Each linear transformation has its own unique matrix. 🤔

• Conversely, if you have a 2x2 matrix A, the function L(v) = Av is always a linear transformation from R² to R². This is because it preserves vector addition and scalar multiplication: L(u + v) = A(u + v) = Au + Av = L(u) + L(v) and L(cu) = A(cu) = c(Au) = cL(u). ☑️

Source: SlideServe

#✖️ Matrix Multiplication

When you multiply a 2x2 transformation matrix A by a 2xn matrix of n input vectors X, you get a 2xn matrix of the n output vectors for L(v) = Av. This works because matrix multiplication follows the distributive property: A(BC) = (AB)C. 🔤

If A = [a11, a12; a21, a22] and X = [v1, v2, ..., vn], then:

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

Each entry in the resulting 2xn matrix is the output vector of L(v) = Av for the corresponding input vector v in X. The output matrix has the same number of columns as the input, but each column is the transformed version of the original vector. 💯

#Final Exam Focus

Okay, let's focus on what's most important for the exam. Here’s what you should absolutely nail down:

• Key Concepts: Linear transformations, transformation matrices, vector representations (zero, single, and multiple vectors), and matrix multiplication.
• High-Priority Topics: Understanding how matrices represent linear transformations and performing matrix multiplication.
• Common Question Types: Applying transformation matrices to vectors, identifying linear transformations, and interpreting matrix results.

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. Which of the following is NOT a property of a linear transformation T? (A) T(u + v) = T(u) + T(v) (B) T(cu) = cT(u) (C) T(0) = 0 (D) T(uv) = T(u)T(v)

2. Given a transformation matrix A = [[2, 1], [1, 3]], what is the result of applying this transformation to the vector v = [1, 2]? (A) [3, 7] (B) [4, 5] (C) [5, 7] (D) [7, 3]

3. A linear transformation T maps the vector [1, 0] to [2, 3] and the vector [0, 1] to [-1, 1]. What is the transformation matrix A? (A) [[2, -1], [3, 1]] (B) [[2, 3], [-1, 1]] (C) [[1, 0], [0, 1]] (D) [[3, 2], [1, -1]]

#Free Response Question

Consider the linear transformation T(x) = Ax, where A = [[1, 2], [-1, 1]].

(a) Find the image of the vector v = [3, -2] under the transformation T. (b) Find the matrix that represents the transformation that first reflects a vector across the x-axis, then applies T. (c) If the transformation T is applied to a set of vectors that form a square with vertices at [1,1], [2,1], [2,2], and [1,2], what is the area of the transformed shape?

Scoring Breakdown:

(a) 2 points - 1 point for correct matrix multiplication - 1 point for the correct image vector

(b) 3 points - 1 point for the correct reflection matrix - 1 point for correct matrix multiplication order - 1 point for the correct combined transformation matrix

(c) 4 points - 1 point for recognizing the transformed shape is a parallelogram (or another shape with equivalent area calculation) - 1 point for finding the transformed vectors of the square's vertices - 1 point for finding the area of the transformed shape using the determinant or base and height - 1 point for the correct numerical answer

Let's get you ready for the exam! You've got this! 💪