Finding a least-squares fit.Linear algebra Find the least-squares line given the matrix products $ A^T A = \begin{bmatrix} 4 & 70 \\ 70 & 1510 \end{bmatrix} $ and $ A^T \mathbf{b} = \begin{bmatrix} 354.4 \\ 6801.8 \end{bmatrix} $.\[ y = \text{Ex: } 1.2 + \text{Ex: } 1.2 \cdot x \]

For a system: Ax=b, the formula for least-squares fit is x^=ATA-1ATb. We know that: a b c d…

Answered: = Find the least-squares line given the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Finding a least-squares fit.

Linear algebra

Find the least-squares line given the matrix products $ A^T A = \begin{bmatrix} 4 & 70 \\ 70 & 1510 \end{bmatrix} $ and $ A^T \mathbf{b} = \begin{bmatrix} 354.4 \\ 6801.8 \end{bmatrix} $.

\[ y = \text{Ex: } 1.2 + \text{Ex: } 1.2 \cdot x \]

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

Expert Solution

Step 1

For a system: $A x = b$ , the formula for least-squares fit is $\hat{x} = {(A^{T} A)}^{- 1} A^{T} b$ .

We know that: ${(\begin{matrix} a & b \\ c & d \end{matrix})}^{- 1} = \frac{1}{\det (\begin{matrix} a & b \\ c & d \end{matrix})} (\begin{matrix} d & - b \\ - c & a \end{matrix})$ .