Change of coordinates matrices.linear algebra 95B == {[] []} and = {[%] [4]}C-Complete the augmented matrix used to find the change of coordinates matrix PC-BEx: 3are bases for R².ŵ

Answered: B = {[3]-[]} and C = {[%] [4]} Ex: 3 2…

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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Related questions

Question

Change of coordinates matrices.

linear algebra

9
5
B =
= {[] []} and = {[%] [4]}
C
-
Complete the augmented matrix used to find the change of coordinates matrix P
C-B
Ex: 3
are bases for R².
ŵ

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

$B = \{[\begin{matrix} 7 \\ 8 \end{matrix}], [\begin{matrix} - 1 \\ - 3 \end{matrix}]\}$ and $C = \{[\begin{matrix} 9 \\ - 6 \end{matrix}], [\begin{matrix} 5 \\ - 7 \end{matrix}]\}$ are bases for $ℝ^{2}$

complete the augmented matrix used to find the change of coordinates matrix $\underset{C \leftarrow B}{P}$

solution

$\underset{B \to C}{P} = [{[b_{1}]}_{C}, {[b_{2}]}_{C}]$

${[b_{1}]}_{C} = [\begin{matrix} 7 \\ 8 \end{matrix}] = x [\begin{matrix} 9 \\ - 6 \end{matrix}] + y [\begin{matrix} 5 \\ - 7 \end{matrix}] augmented matrix \Rightarrow [\begin{matrix} 9 & 5 & 7 \\ - 6 & - 7 & 8 \end{matrix}] R_{1} \to \frac{R_{1}}{9} [\begin{matrix} 1 & \frac{5}{9} & \frac{7}{9} \\ - 6 & - 7 & 8 \end{matrix}]$