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Let $a$ and $b$ be nonzero numbers. Show that the mapping $T$ defined by $T\left\{y_{k}\right\}=\left\{w_{k}\right\},$ where$w_{k}=y_{k+2}+a y_{k+1}+b y_{k}$is a linear transformation from $\mathrm{S}$ into $\mathrm{S}$

the mapping $T$ is linear transformation.

Calculus 3

Chapter 4

Vector Spaces

Section 8

Applications to Difference Equations

Vectors

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So, first of all observer that he told the one is a transformation from M in our two women are there is it takes. And anyway, in Matrix and our ports and in very in matrix and you're given to you to anyone individually So we just have to compute. What happens for each of them interests is so t two t one off is just even off a and then you applied took even if it so he won off is just a minus a transpose. So this is summon bay in metrics and they do off a matrix is a matrix place It's trance fools. So you take this and you add X transpose No transport off some of two matrices. It's some off transports off individual metrics so you can open up records like this. A transport a V minus eight transports is just gay transpose minus transport off transports, which is just e. Okay, now, as you can see, this is zero. So no matter what metrics, what matrix You input into t 2 50 when he will get zero. So therefore Kato off even is the zero transformation

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