Problem 1 - Calculus for Biology and Medicine

Question

Let $$ A=\left[\begin{array}{rr} 2 & 2 \\ -1 & 4 \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right], \quad \text { and } \quad \mathbf{y}=\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right] $$ (a) Show by direct calculation that \(A(\mathbf{x}+\mathbf{y})=A \mathbf{x}+A \mathbf{y}\). (b) Show by direct calculation that \(A(\lambda \mathbf{x})=\lambda(A \mathbf{x})\).

Step-by-Step Solution

Answer

For part (a), specifically, we showed step-by-step that adding first results in the same outcome as adding matrix products. For part (b), we demonstrated that scaling first produces the same result as scaling the matrix products.

Step 1: Setting Up the Expression for Part (a)

We need to show that \(A(\mathbf{x}+\mathbf{y}) = A\mathbf{x} + A\mathbf{y}\). First, calculate \(\mathbf{x} + \mathbf{y}\):\[\mathbf{x} + \mathbf{y} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} y_1 \ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix}.\]

Step 2: Apply Matrix A to (x+y)

Now, compute \(A(\mathbf{x} + \mathbf{y})\). Multiply matrix \(A\) by the vector \(\mathbf{x} + \mathbf{y}\):\[A(\mathbf{x}+\mathbf{y}) = \begin{bmatrix} 2 & 2 \ -1 & 4 \end{bmatrix} \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix}.\]This results in:\[= \begin{bmatrix} 2(x_1 + y_1) + 2(x_2 + y_2) \ -1(x_1 + y_1) + 4(x_2+ y_2) \end{bmatrix} = \begin{bmatrix} 2x_1 + 2y_1 + 2x_2 + 2y_2 \ -x_1 - y_1 + 4x_2 + 4y_2 \end{bmatrix}.\]

Step 3: Calculate A(x) and A(y) Separately

First, calculate \(A\mathbf{x}\):\[A\mathbf{x} = \begin{bmatrix} 2 & 2 \ -1 & 4 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 2x_1 + 2x_2 \ -x_1 + 4x_2 \end{bmatrix}.\]Now calculate \(A\mathbf{y}\):\[A\mathbf{y} = \begin{bmatrix} 2 & 2 \ -1 & 4 \end{bmatrix} \begin{bmatrix} y_1 \ y_2 \end{bmatrix} = \begin{bmatrix} 2y_1 + 2y_2 \ -y_1 + 4y_2 \end{bmatrix}.\]

Step 4: Add A(x) and A(y) to Verify Equality

Add \(A\mathbf{x}\) and \(A\mathbf{y}\):\[A\mathbf{x} + A\mathbf{y} = \begin{bmatrix} 2x_1 + 2x_2 \ -x_1 + 4x_2 \end{bmatrix} + \begin{bmatrix} 2y_1 + 2y_2 \ -y_1 + 4y_2 \end{bmatrix} = \begin{bmatrix} 2x_1 + 2y_1 + 2x_2 + 2y_2 \ -x_1 - y_1 + 4x_2 + 4y_2 \end{bmatrix}.\]This is equal to \(A(\mathbf{x} + \mathbf{y})\) from Step 2, confirming the equality.

Step 5: Setting Up the Expression for Part (b)

For part (b), we need to show that \(A(\lambda \mathbf{x}) = \lambda(A \mathbf{x})\). Calculate \(\lambda \mathbf{x}\):\[\lambda \mathbf{x} = \lambda \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix}.\]

Step 6: Apply Matrix A to (λx)

Compute \(A(\lambda \mathbf{x})\) by applying matrix \(A\) to \(\lambda \mathbf{x}\):\[A(\lambda \mathbf{x}) = \begin{bmatrix} 2 & 2 \ -1 & 4 \end{bmatrix} \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix} = \begin{bmatrix} 2(\lambda x_1) + 2(\lambda x_2) \ -1(\lambda x_1) + 4(\lambda x_2) \end{bmatrix}.\]This simplifies to:\[= \begin{bmatrix} \lambda(2x_1 + 2x_2) \ \lambda(-x_1 + 4x_2) \end{bmatrix}.\]

Step 7: Calculate λ(Ax)

Calculate \(\lambda(A\mathbf{x})\) based on \(A\mathbf{x}\) from Step 3:\[\lambda(A\mathbf{x}) = \lambda \begin{bmatrix} 2x_1 + 2x_2 \ -x_1 + 4x_2 \end{bmatrix} = \begin{bmatrix} \lambda(2x_1 + 2x_2) \ \lambda(-x_1 + 4x_2) \end{bmatrix}.\]This matches \(A(\lambda \mathbf{x})\) from Step 6, confirming the second property.

Key Concepts

Matrix Addition

Matrix addition is the process of summing two matrices by adding their corresponding elements. This operation is only defined when the matrices involved are of the same dimensions. To perform matrix addition:

• Align matrices of the same size.
• Add each element from one matrix to the corresponding element in the other matrix.

For example, consider two matrices \( ext{Matrix } M ext{ and Matrix } N \) both of size \( 2 \times 2 \):\[M = \begin{bmatrix} m_{11} & m_{12} \ m_{21} & m_{22} \end{bmatrix}, \quad N = \begin{bmatrix} n_{11} & n_{12} \ n_{21} & n_{22} \end{bmatrix}\]Their sum \( M + N \) will be:\[M + N = \begin{bmatrix} m_{11} + n_{11} & m_{12} + n_{12} \ m_{21} + n_{21} & m_{22} + n_{22} \end{bmatrix}\]When you add the vectors \( \mathbf{x} \text{ and } \mathbf{y} \), as shown in the exercise, you follow the same idea of adding corresponding elements.Understanding this operation is pivotal because matrix addition is fundamental in vector spaces and in forming linear combinations.

Scalar Multiplication

Scalar multiplication in the context of matrices or vectors involves multiplying every entry of a matrix or a vector by a scalar (a single number). This operation is crucial as it assists in scaling transformations, which are common in various applications.

• The matrix or vector remains of the same size after the operation.
• Each element is simply multiplied by the scalar to produce a new matrix or vector.

For instance, if you have a vector \( \mathbf{v} = \begin{bmatrix} v_1 \ v_2 \end{bmatrix} \) and a scalar \( \lambda \), the result of \( \lambda \mathbf{v} \) is:\[\lambda \mathbf{v} = \begin{bmatrix} \lambda v_1 \ \lambda v_2 \end{bmatrix}\]In the exercise, you demonstrated that applying a matrix \( A \) to a scaled vector \( \lambda \mathbf{x} \) yields the same result as scaling the result of applying \( A \) to \( \mathbf{x} \) by \( \lambda \). This associative property is a key feature of linear transformations.

Vector Spaces

A vector space is a collection of vectors that can be added together and multiplied by scalars, following certain rules. They are foundational in the study of linear algebra and appear in many other mathematical disciplines. Here are some essential properties of vector spaces:

• Closure under addition: The sum of any two vectors in a vector space is also in the space.
• Closure under scalar multiplication: Multiplying a vector by a scalar yields another vector in the space.
• Existence of an additive identity: There is a zero vector in the space, which does not affect other vectors when added.
• Commutativity and associativity: Vector addition is both commutative (order doesn't matter) and associative (grouping doesn't matter).

In the exercise scenario, the operations you performed showcase the principles of vector spaces. Matrix application preserves operations like addition \( (A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y}) \) and scalar multiplication \((A(\lambda \mathbf{x}) = \lambda (A\mathbf{x}))\), which mean the set of possible outputs forms a vector space. Knowing these characteristics helps in recognizing the structure and properties of various mathematical systems.