Problem 2 - Calculus for Biology and Medicine

Question

Let $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right], \quad \text { and } \quad \mathbf{y}=\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right] $$ (a) Show by direct calculation that \(A(\mathbf{x}+\mathbf{y})=A \mathbf{x}+A \mathbf{y}\). $$ \text { (b) Show by direct calculation that } A(\lambda \mathbf{x})=\lambda(A \mathbf{x}) \text { . } $$

Step-by-Step Solution

Answer

(a) \( A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y} \); (b) \( A(\lambda \mathbf{x}) = \lambda(A \mathbf{x}) \).

Step 1: Declare Matrices and Vectors

We are given a matrix \( A \) and vectors \( \mathbf{x} \) and \( \mathbf{y} \): \[ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}, \quad \mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix}. \] Our task is to prove two conditions showing properties of linear transformations using direct calculation.

Step 2: Calculate \( \mathbf{x} + \mathbf{y} \)

Compute the sum of vectors \( \mathbf{x} \) and \( \mathbf{y} \): \[ \mathbf{x} + \mathbf{y} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} y_1 \ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix}. \]

Step 3: Multiply \( A \) by \( \mathbf{x} + \mathbf{y} \)

Use matrix multiplication to compute \( A(\mathbf{x} + \mathbf{y}) \): \[ A(\mathbf{x} + \mathbf{y}) = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix} = \begin{bmatrix} a_{11}(x_1 + y_1) + a_{12}(x_2 + y_2) \ a_{21}(x_1 + y_1) + a_{22}(x_2 + y_2) \end{bmatrix}. \]

Step 4: Expand Matrix Multiplication Result

Expand the terms in the result: \[ A(\mathbf{x} + \mathbf{y}) = \begin{bmatrix} a_{11}x_1 + a_{11}y_1 + a_{12}x_2 + a_{12}y_2 \ a_{21}x_1 + a_{21}y_1 + a_{22}x_2 + a_{22}y_2 \end{bmatrix}. \]

Step 5: Calculate \( A \mathbf{x} \) and \( A \mathbf{y} \) Separately

Compute each matrix-vector multiplication separately: \[ A \mathbf{x} = \begin{bmatrix} a_{11}x_1 + a_{12}x_2 \ a_{21}x_1 + a_{22}x_2 \end{bmatrix}, \quad A \mathbf{y} = \begin{bmatrix} a_{11}y_1 + a_{12}y_2 \ a_{21}y_1 + a_{22}y_2 \end{bmatrix}. \]

Step 6: Add \( A \mathbf{x} \) and \( A \mathbf{y} \) Together

Add the results of \( A \mathbf{x} \) and \( A \mathbf{y} \): \[ A \mathbf{x} + A \mathbf{y} = \begin{bmatrix} a_{11}x_1 + a_{12}x_2 \ a_{21}x_1 + a_{22}x_2 \end{bmatrix} + \begin{bmatrix} a_{11}y_1 + a_{12}y_2 \ a_{21}y_1 + a_{22}y_2 \end{bmatrix} = \begin{bmatrix} a_{11}x_1 + a_{11}y_1 + a_{12}x_2 + a_{12}y_2 \ a_{21}x_1 + a_{21}y_1 + a_{22}x_2 + a_{22}y_2 \end{bmatrix}. \]

Step 7: Conclude Part (a)

Since \( A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y} \) holds true from direct calculations, we have shown that the linearity properties are satisfied for addition.

Step 8: Calculate \( \lambda \mathbf{x} \)

Compute the scalar multiplication of vector \( \mathbf{x} \): \[ \lambda \mathbf{x} = \lambda \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix}. \]

Step 9: Multiply \( A \) by \( \lambda \mathbf{x} \)

Compute \( A(\lambda \mathbf{x}) \) using matrix multiplication: \[ A(\lambda \mathbf{x}) = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix} = \begin{bmatrix} a_{11}(\lambda x_1) + a_{12}(\lambda x_2) \ a_{21}(\lambda x_1) + a_{22}(\lambda x_2) \end{bmatrix}. \]

Step 10: Simplify \( A(\lambda \mathbf{x}) \)

Factor \( \lambda \) out of each term: \[ A(\lambda \mathbf{x}) = \begin{bmatrix} \lambda (a_{11} x_1) + \lambda (a_{12} x_2) \ \lambda (a_{21} x_1) + \lambda (a_{22} x_2) \end{bmatrix} = \lambda \begin{bmatrix} a_{11} x_1 + a_{12} x_2 \ a_{21} x_1 + a_{22} x_2 \end{bmatrix}. \]

Step 11: Calculate \( \lambda(A \mathbf{x}) \)

Compute \( \lambda(A \mathbf{x}) \) by scaling \( A \mathbf{x} \): \[ \lambda(A \mathbf{x}) = \lambda \begin{bmatrix} a_{11} x_1 + a_{12} x_2 \ a_{21} x_1 + a_{22} x_2 \end{bmatrix} = \begin{bmatrix} \lambda(a_{11} x_1 + a_{12} x_2) \ \lambda(a_{21} x_1 + a_{22} x_2) \end{bmatrix}. \]

Step 12: Conclude Part (b)

Since \( A(\lambda \mathbf{x}) = \lambda(A \mathbf{x}) \) holds true from the calculations, this demonstrates that scalar multiplication follows linearity with respect to the matrix transformation.

Key Concepts

Matrix Multiplication

In linear algebra, matrix multiplication is a fundamental operation, crucial for transforming vectors and solving systems of equations. To multiply a matrix by a vector, you multiply each element of a row of the matrix by the corresponding element of the column vector, and then sum those products.
For example, if you have:

• Matrix: \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)
• Vector: \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \)

Matrix-vector multiplication \( A \mathbf{x} \) is performed as follows:\[A \mathbf{x} = \begin{bmatrix} a_{11}x_1 + a_{12}x_2 \a_{21}x_1 + a_{22}x_2 \end{bmatrix}\]This operation is transformational, meaning it changes the vector \( \mathbf{x} \) into a new vector in the same or a different space. Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the vector it multiplies.

Vector Addition

Vector addition involves adding two vectors component-wise to produce a new vector. This operation is straightforward and is often used in physics and engineering to combine forces or other quantities represented as vectors.

For vectors \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \), addition is performed as follows:

• Add corresponding components: \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix} \)

Vector addition follows the commutative property, meaning that \( \mathbf{x} + \mathbf{y} = \mathbf{y} + \mathbf{x} \). It is also associative, so \( (\mathbf{x} + \mathbf{y}) + \mathbf{z} = \mathbf{x} + (\mathbf{y} + \mathbf{z}) \), which helps simplify complex operations involving multiple vectors.

Scalar Multiplication

Scalar multiplication in linear algebra involves multiplying a vector by a scalar (a real number), changing the magnitude of the vector without altering its direction. When you multiply each component of a vector by the same scalar, the result is a scaled version of the original vector.

For a vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) and a scalar \( \lambda \), the multiplication is performed as follows:

• Multiply each component by the scalar: \( \lambda \mathbf{x} = \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix} \)

Scalar multiplication adheres to the distributive property, which is formalized as \( \lambda(\mathbf{x} + \mathbf{y}) = \lambda \mathbf{x} + \lambda \mathbf{y} \). This property plays a crucial role in understanding the linearity of transformations in vector spaces.

Linearity Properties

Linearity properties form the backbone of linear transformations. These properties ensure transformations are predictable and consistent across operations.

Two key linearity properties in linear transformations include:

• Additivity: For a matrix \( A \), and vectors \( \mathbf{x} \), \( \mathbf{y} \), additivity is expressed as \( A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y} \). This means the transform of a sum is the sum of the transforms.
• Homogeneity: For a matrix \( A \), vector \( \mathbf{x} \), and scalar \( \lambda \), this is expressed as \( A(\lambda \mathbf{x}) = \lambda(A\mathbf{x}) \). This implies that scaling a vector before or after applying the matrix results in the same scaled transform.

Together, these properties demonstrate that certain operations like addition or scaling retain linear relationships. They are why many problems in engineering, graphics, and physics prefer linear models which are simpler to compute and predict.