Q3.1-14E For each matrix A聽in exercises ... [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: Q3.1-14E (page 119)

For each matrix $A$ in exercises 14 through 16, find vectors that span the image of $A$ . Give as few vectors as possible. Use paper and pencil.
$A = [\begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{matrix}]$

Short Answer

Expert verified

The image of $A$ is role="math" localid="1664170177681" $im (A) = s p a n ([\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}])$ .

Step by step solution

Determine the image of a matrix

The image of a matrix $A$ is the span of the column vectors of $A$ .

Find kernel of the given matrix

The image of a matrix $A$ consists of all vectors $\overset{鈫�}{x}$ which span the column vectors of $A$ . So,

$[\begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = x_{1} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] + x_{2} [\begin{matrix} 2 \\ 2 \\ 2 \end{matrix}] + x_{3} [\begin{matrix} 3 \\ 3 \\ 3 \end{matrix}] = x_{1} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] + 2 x_{2} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] + 3 x_{3} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = (x_{1} + 2 x_{2} + 3 x_{3}) [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$

So, the vector $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ spans the column vectors of $A$ .

Thus, $im (A) = s p a n ([\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}])$ .

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91影视

For each matrix $A$ in exercises 14 through 16, find vectors that span the image of $A$ . Give as few vectors as possible. Use paper and pencil.
$A = [\begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{matrix}]$

Short Answer

Step by step solution

Determine the image of a matrix

Find kernel of the given matrix

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91影视

For each matrix Ain exercises 14 through 16, find vectors that span the image of A. Give as few vectors as possible. Use paper and pencil.A=[123123123]

Short Answer

Step by step solution

Determine the image of a matrix

Find kernel of the given matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Calculus

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

For each matrix $A$ in exercises 14 through 16, find vectors that span the image of $A$ . Give as few vectors as possible. Use paper and pencil.
$A = [\begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{matrix}]$