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Q33E

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Found in: Page 120

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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# Give an example of a linear transformation whose kernel is the plane ${\mathbit{x}}{\mathbf{+}}{\mathbf{2}}{\mathbit{y}}{\mathbf{+}}{\mathbf{3}}{\mathbit{z}}{\mathbf{=}}{\mathbf{0}}$in ${{\mathbf{ℝ}}}^{{\mathbf{3}}}$.

The required linear transformation is,

$A\stackrel{\to }{v}=\left(x+2y+3z, 0, 0\right)$

See the step by step solution

## Step by Step Solution:  Step 1: To define kernel of linear transformation

The kernel of linear transformation is defined as follows:

The kernel of a linear transformation $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ from ${\mathrm{ℝ}}^{m}$ to ${\mathrm{ℝ}}^{n}$ consists of all zeros of the transformation, i.e., the solutions of the equations $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}=0$

It is denoted by ker$\left(T\right)$or ker $\left(A\right)$

## Step 2: To give an example of a linear transformation

We require linear transformation $A$such that

$A\stackrel{\to }{v}=0$,where the plane is $\stackrel{\to }{v}=\left(x, y, z\right): x+2y+3z=0$is ${\mathrm{ℝ}}^{3}$is the kernel of the transformation.

Since we know that the dot product of the required vector with normal vector of given plane given by,

$\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$,is equal to 0, the required linear transformation is,

$A\stackrel{\to }{v}=\left[\begin{array}{ccc}1& 2& 3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]·\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A\stackrel{\to }{v}=\left(x+2y+3z, 0, 0\right)$

where, $\stackrel{\to }{v}=\left(x,y,z\right): x+2y+3z=0$

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