Q34E Give an example of a linear tran... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q34E (page 120)

Give an example of a linear transformation whose kernel is the line spanned by $[\begin{matrix} - 1 \\ 1 \\ 2 \end{matrix}]$ in ${鈩�}^{3}$

Short Answer

Expert verified

The required linear transformation is, $A = [\begin{matrix} 1 & 1 & 0 \\ - 1 & 1 & - 1 \\ 0 & 0 & 0 \end{matrix}]$ .

Step by step solution

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

Thekernel of linear transformationis defined as follows:

The kernel of a linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ from ${鈩�}^{m}$ to ${鈩�}^{n}$ consists of all zeros of the transformation, i.e., the solutions of the equations $T (\overset{鈫�}{x}) = A \overset{鈫�}{x} = 0$

It is denoted by ker $(T)$ or ker $(A)$ .

The given vector is, $[\begin{matrix} - 1 \\ 1 \\ 2 \end{matrix}]$ in ${鈩�}^{3}$ ,which is the plane $- x + y + 2 z = 0$

To give an example of a linear transformation

Consider the vectors ${\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}]$ and ${\overset{鈫�}{v}}_{2} = [\begin{matrix} - 1 \\ 1 \\ - 1 \end{matrix}]$

These vectors are non-collinear.

Therefore, it can be represented in the matrix form as:

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

where, the rows of matrix are the above vectors, which is the required linear transformation.

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91影视

Give an example of a linear transformation whose kernel is the line spanned by $[\begin{matrix} - 1 \\ 1 \\ 2 \end{matrix}]$ in ${鈩�}^{3}$

Short Answer

Step by step solution

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

To give an example of a linear transformation

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91影视

Give an example of a linear transformation whose kernel is the line spanned by [-112]in鈩�3

Short Answer

Step by step solution

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

To give an example of a linear transformation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Give an example of a linear transformation whose kernel is the line spanned by $[\begin{matrix} - 1 \\ 1 \\ 2 \end{matrix}]$ in ${鈩�}^{3}$