Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Express the line $L$ in $ℝ^{3}$ spanned by the vector $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ as the image of a matrix $A$ and as the kernel of a matrix $B$ .

Matrix $A$

$A = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$

Matrix $B$

$B = [\begin{matrix} 1 & 1 & - 2 \\ 1 & - 1 & 0 \end{matrix}]$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Spanned by vector

For a line to be an image of a matrix $A$ it has to be spanned by column vectors of $A$ , and since it is spanned by a vector already we'll use it as a column vector

$A = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$

For a line to be a kernel of matrix $B$ the equality must be correct

$B = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = 0$

Step 2: Choose vector that gives 0 in dot product

Since we have no other conditions we can choose any vector that gives 0 in dot product with

$\vec{v} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$

We choose vector

$\vec{w} = [\begin{matrix} 1 \\ 1 \\ - 2 \end{matrix}]$

But we can construct another vector that gives 0 in dot product with $\vec{w}$ so our matrix is not complete, we know that kernel of must form a plane.

So we find another vector $\vec{k}$ that is non collinear with $\vec{w}$ and it's dot product with $\vec{v}$ is 0 let's say

$\vec{k} = [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}]$

Step 3: Construct matrix B

Now we have two non collinear vectors so they span a plane and now we construct matrix $B$

$B = [\begin{matrix} 1 & 1 & - 2 \\ 1 & - 1 & 0 \end{matrix}]$

Step 4: The final answer

Matrix $A$

$A = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$

Matrix $B$

$B = [\begin{matrix} 1 & 1 & - 2 \\ 1 & - 1 & 0 \end{matrix}]$

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Linear Algebra With Applications

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Short Answer

Express the line $L$ in $ℝ^{3}$ spanned by the vector $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ as the image of a matrix $A$ and as the kernel of a matrix $B$ .

Step by Step Solution

Step 1: Spanned by vector

Step 2: Choose vector that gives 0 in dot product

Step 3: Construct matrix B

Step 4: The final answer

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Express the line L in ℝ3 spanned by the vector [111] as the image of a matrix Aand as the kernel of a matrix B.

Step by Step Solution

Step 1: Spanned by vector

Step 2: Choose vector that gives 0 in dot product

Step 3: Construct matrix B

Step 4: The final answer

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Express the line $L$ in $ℝ^{3}$ spanned by the vector $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ as the image of a matrix $A$ and as the kernel of a matrix $B$ .