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 \(A(\mathbf{x}+\mathbf{y})=A \mathbf{x}+A \mathbf{y}\). (b) Show by direct calculation that \(A(\lambda \mathbf{x})=\lambda(A \mathbf{x})\).

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: Add the Two Vectors

We are tasked with proving that the sum of the transformed vectors is equivalent to the transformation of the sum of the vectors. We begin by summing the vectors \( \mathbf{x} \) and \( \mathbf{y} \), resulting in: \[ \mathbf{x} + \mathbf{y} = \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix}. \]

Step 2: Multiply the Matrix by the Sum of Vectors

Next, multiply matrix \( A \) by the resulting vector sum: \[ A(\mathbf{x} + \mathbf{y}) = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix}.\] This gives us: \[ \begin{bmatrix} a_{11}(x_1 + y_1) + a_{12}(x_2 + y_2) \ a_{21}(x_1 + y_1) + a_{22}(x_2 + y_2) \end{bmatrix}. \]

Step 3: Distribute Multiplication Over Addition

Distribute the matrix multiplication over the added vector components: \[ = \begin{bmatrix} a_{11}x_1 + a_{11}y_1 + a_{12}x_2 + a_{12}y_2 \ a_{21}x_1 + a_{21}y_1 + a_{22}x_2 + a_{22}y_2 \end{bmatrix}. \]

Step 4: Separate Terms to Match Individual Vector Transformations

Rearrange and separate terms to match the effect of the matrix acting on the vectors individually: \[ = \begin{bmatrix} (a_{11}x_1 + a_{12}x_2) + (a_{11}y_1 + a_{12}y_2) \ (a_{21}x_1 + a_{22}x_2) + (a_{21}y_1 + a_{22}y_2) \end{bmatrix}. \]

Step 5: Compute Matrix-Vector Product for Each Vector

Calculate \( A\mathbf{x} \) and \( A\mathbf{y} \) 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 the Results of the Two Matrix-Vector Products

Sum the results from Step 5: \[ A\mathbf{x} + A\mathbf{y} = \begin{bmatrix} (a_{11}x_1 + a_{12}x_2) + (a_{11}y_1 + a_{12}y_2) \ (a_{21}x_1 + a_{22}x_2) + (a_{21}y_1 + a_{22}y_2) \end{bmatrix}. \]

Step 7: Conclude Part (a)

This verifies our requirement: \( A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y} \).

Step 8: Scale the Vector \(\mathbf{x}\) by \(\lambda\)

Now, for part (b), calculate \( \lambda \mathbf{x} = \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix}. \)

Step 9: Multiply the Matrix by the Scaled Vector

Multiply the matrix \( A \) by the scaled vector: \[ A(\lambda \mathbf{x}) = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix}, \] resulting in \[ \begin{bmatrix} a_{11}(\lambda x_1) + a_{12}(\lambda x_2) \ a_{21}(\lambda x_1) + a_{22}(\lambda x_2) \end{bmatrix}. \]

Step 10: Factor \(\lambda\) Out of the Result

Factor \( \lambda \) out from the expression: \[ = \lambda \begin{bmatrix} a_{11}x_1 + a_{12}x_2 \ a_{21}x_1 + a_{22}x_2 \end{bmatrix}. \]

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

Observe that this expression is equivalent to: \( \lambda (A \mathbf{x}) \). Hence, \( A(\lambda \mathbf{x}) = \lambda (A \mathbf{x}) \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It describes how we multiply a matrix by another matrix or a vector. It's not as straightforward as multiplying two numbers. When a matrix multiplies a vector, it transforms or alters the vector according to the rules encoded in the matrix. This operation involves combining the rows of the matrix with the columns of the vector, usually resulting in another vector.

In our exercise, we have:

• Matrix \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)
• Vectors \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \)

To execute matrix multiplication, you take each row of matrix \( A \) and perform the dot product with vector \( \mathbf{x} \). For example, the first component of the resulting vector is calculated as \( a_{11}x_1 + a_{12}x_2 \). This process is repeated for each component. It's important to remember that the number of columns in the matrix must match the number of elements in the vector for the multiplication to be valid.

Vector Addition

Vector addition is another key operation in linear algebra. It is all about combining two vectors to form a new vector. In this process, each component of one vector is added to the corresponding component of another vector.

For instance, if we consider vectors \( \mathbf{x} \) and \( \mathbf{y} \):

• \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \)
• \( \mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \)

Adding these vectors gives us a new vector:\[ \mathbf{x} + \mathbf{y} = \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix} \]This operation is straightforward and element-wise. It's crucial to ensure both vectors have the same dimensions for the addition to work. This concept is essential in transformations, where these combined effects often simplify our calculations. In our case, we verified how matrix multiplication distributes over the addition of vectors.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number), resulting in a vector that is scaled by that number. This operation affects every component of the vector equally.

Consider a scalar \( \lambda \) and vector \( \mathbf{x} \):

• \( \lambda \in \mathbb{R} \) (real number)
• \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \)

The result of scalar multiplication is:\[ \lambda \mathbf{x} = \begin{bmatrix} \lambda x_1 \ \lambda x_2 \end{bmatrix} \]Each element of the vector is multiplied by the scalar. This operation is helpful for scaling vectors in various applications, such as physics (e.g., changing the magnitude of forces) and engineering. In our exercise, it helped demonstrate how a matrix scales the entire transformation just like it does each individual vector.

Distributive Property

The distributive property is a fundamental algebraic principle allowing us to expand expressions in a way that simplifies calculations. It states that multiplying a sum by a number (or another algebraic entity) is the same as doing each multiplication separately and then adding the results. Mathematically, it says:\[ a(b + c) = ab + ac \]This property holds in both arithmetic and algebra, including for matrices and vector operations.

In our exercise, the distributive property allowed us to show:

• How matrix \( A \) acting on the sum of vectors \( \mathbf{x} + \mathbf{y} \) produces the same result as \( A \) acting on each vector individually and then summing the results.
• For a scalar \( \lambda \) and vector \( \mathbf{x} \), how \( A(\lambda \mathbf{x}) \) simplifies to \( \lambda (A \mathbf{x}) \).

Understanding the distributive property helps in verifying and simplifying complex algebraic manipulations in linear algebra, ensuring the synthesis of concepts through straightforward arithmetic operations.