Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q35E (page 36)

In ${鈩�}^{m}$ , we define
$\overset{鈫�}{e_{i}} = [\begin{matrix} 0 \\ 0 \\ 鈰� \\ 1 \\ 鈰� \\ 0 \end{matrix}] 鈫� i^{t h} c o m p o n e n t$
If A is an $n 脳 m$ matrix, what is $A \overset{鈫�}{e_{i}}$ ?

Short Answer

Expert verified

The $i^{t h}$ column of A is $A \overset{鈫�}{e_{i}}$ .

Step by step solution

Consider the matrices.

If A is an n x m matrix with row vectors $\overset{鈫�}{蝇_{1},} . . . . . . . \overset{鈫�}{, 蝇_{n}}$ and $\overset{鈫�}{x}$ is a vector in $R^{m}$ then,

$A \overset{鈫�}{x} = [\begin{matrix} - {\overset{鈫�}{蝇}}_{1} - \\ 鈰� \\ - \overset{鈫�}{蝇_{n}} - \end{matrix}] \overset{鈫�}{x} [\begin{matrix} - {\overset{鈫�}{蝇}}_{1} . \overset{鈫�}{x} - \\ 鈰� \\ - \overset{鈫�}{蝇_{n}} . \overset{鈫�}{x} - \end{matrix}]$

The matrices are,

$A = [\begin{matrix} a_{11} & 鈥� & a_{1 m} \\ 鈰� & 鈰� & 鈰� \\ a_{n 1} & 鈥� & a_{n m} \end{matrix}] \overset{鈫�}{e_{i}} = [\begin{matrix} 0 \\ 0 \\ 鈰� \\ 1 \\ 鈰� \\ 0 \end{matrix}] 鈫� 鈥� 鈥� 鈥� 鈥� i^{th} component$

A is the n x m matrix.

Perform the multiplication operation

The $i^{t h}$ component of $\overset{鈫�}{e_{i}}$ is 1 and the rest of all the elements are zero. The matrix A has all nonzero elements.

$A \overset{鈫�}{e}$ is basically a multiplication operation of matrices. As all the elements of $\overset{鈫�}{e}$ are zero except the $i^{t h}$ component, thus, the role="math" localid="1664205251359" $A \overset{鈫�}{e_{1}}$ will be the $i^{t h}$ column of A.

Hence, $A \overset{鈫�}{e_{1}}$ is the $i^{t h}$ column of A.

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91影视

In ${鈩�}^{m}$ , we define
$\overset{鈫�}{e_{i}} = [\begin{matrix} 0 \\ 0 \\ 鈰� \\ 1 \\ 鈰� \\ 0 \end{matrix}] 鈫� i^{t h} c o m p o n e n t$
If A is an $n 脳 m$ matrix, what is $A \overset{鈫�}{e_{i}}$ ?

Short Answer

Step by step solution

Consider the matrices.

Perform the multiplication operation

One App. One Place for Learning.

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91影视

In 鈩�m, we defineei鈫�=[00鈰�1鈰�0]鈫�ithcomponentIf A is ann脳m matrix, what isAei鈫� ?

Short Answer

Step by step solution

Consider the matrices.

Perform the multiplication operation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Logic and Functions

Pure Maths

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In ${鈩�}^{m}$ , we define
$\overset{鈫�}{e_{i}} = [\begin{matrix} 0 \\ 0 \\ 鈰� \\ 1 \\ 鈰� \\ 0 \end{matrix}] 鈫� i^{t h} c o m p o n e n t$
If A is an $n 脳 m$ matrix, what is $A \overset{鈫�}{e_{i}}$ ?