Q47E Let T be the projection along a ... [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: Q47E (page 121)

Let T be the projection along a line $L_{1}$ onto a line role="math" localid="1660726103462" $L_{2}$ . See Exercise 2.2.33. Describe the image and the kernel of T geometrically.

Short Answer

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Step by step solution

Step 1:Objective

It is given that T is the projection along a line $L_{1}$ onto a line $L_{2}$ . The objective is to describe the image and range of T geometrically.

Explanation

Since T is a projection along a line $L_{1}$ onto a line $L_{2}$ any vector $\overset{鈫�}{x}$ in R虏 can be expressed uniquely as $\overset{鈫�}{x} = {\overset{鈫�}{v}}_{1} + {\overset{鈫�}{v}}_{2}$ where ${\overset{鈫�}{v}}_{1}$ is on the line $L_{1}$ and is on the line L鈧�. The transformation T is given by $T (\overset{鈫�}{x}) = {\overset{鈫�}{v}}_{2}$ Finding the image of T. For every vector $\overset{鈫�}{x}$ in R虏, $T (\overset{鈫�}{x}) = \overset{鈫�}{v_{2}}$ where $\overset{鈫�}{v_{2}}$ is on the line L鈧�.All vectors on the line $L_{2}$ are in the image of T.So, im(T)=L鈧�. Find the kernel of T. Ker(T) consists of all vectors $\overset{鈫�}{x}$ such

that $T (\overset{鈫�}{x}) = \overset{鈫�}{0}$ Every vector $\overset{鈫�}{x}$ on the line $L_{1}$ can be written as $\overset{鈫�}{x} = {\overset{鈫�}{v}}_{1} + \overset{鈫�}{v_{2}}$ where and v鈧�= 0

So for all vectors on the line L鈧�, $T (\overset{鈫�}{x}) = \overset{鈫�}{0}$

So, ker(T) = $L_{1}$ .

Step 3: Describing geometrically

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91影视

Let T be the projection along a line $L_{1}$ onto a line role="math" localid="1660726103462" $L_{2}$ . See Exercise 2.2.33. Describe the image and the kernel of T geometrically.

Short Answer

Step by step solution

Step 1:Objective

Explanation

Step 3: Describing geometrically

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91影视

Let T be the projection along a lineL1 onto a line role="math" localid="1660726103462" L2. See Exercise 2.2.33. Describe the image and the kernel of T geometrically.

Short Answer

Step by step solution

Step 1:Objective

Explanation

Step 3: Describing geometrically

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Let T be the projection along a line $L_{1}$ onto a line role="math" localid="1660726103462" $L_{2}$ . See Exercise 2.2.33. Describe the image and the kernel of T geometrically.