Q3.1-17E For each matrix A聽in Exercise 1... [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: Q3.1-17E (page 119)

For each matrix $A$ in Exercise 17 through 22, describe the image of the transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically (as a line, plane, etc. in ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Short Answer

Expert verified

Thus, the image of $T$ represents R².

Step by step solution

Span of the given matrix.

The given matrix is:

$A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

The objective is to describe the image of the transformation $T (x) = A x$ geometrically.

The image of a linear transformation $T :^{''} 鈫�^{''}$ is

$im (T) = {T (\overset{鈫�}{x}) : \overset{鈫�}{x} \hat{I}^{''}} = {A \overset{鈫�}{x} : \overset{鈫�}{x} \hat{I}^{''}}$

Let localid="1664175933631" $\overset{鈫�}{x} = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] 鈭�^{2}$ . Then the image of the vector localid="1664176580723">x鈫�=x1x2is:

Possible set of vectors.

Let $x = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$ then the equation solve as follows:

$T (x) = A x = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} x_{1} + 2 x_{2} \\ 3 x_{1} + 4 x_{2} \end{matrix}] = x_{1} [\begin{matrix} 1 \\ 3 \end{matrix}] + x_{2} [\begin{matrix} 2 \\ 4 \end{matrix}]$

Thus, the vector $\overset{鈫�}{x} = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] 鈭�^{2}$ in the image of $T$ are spanned by the vectors $[\begin{matrix} 1 \\ 3 \end{matrix}], [\begin{matrix} 2 \\ 4 \end{matrix}]$ .

That means the vectors localid="1664176563286">13,24, span every vector in R².

Hence, the image of $T$ representsR².

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91影视

For each matrix $A$ in Exercise 17 through 22, describe the image of the transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically (as a line, plane, etc. in ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Short Answer

Step by step solution

Span of the given matrix.

Possible set of vectors.

One App. One Place for Learning.

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91影视

For each matrix Ain Exercise 17 through 22, describe the image of the transformation T(x鈫�)=Ax鈫�geometrically (as a line, plane, etc. in 鈩�2and鈩�3.A=[1234]

Short Answer

Step by step solution

Span of the given matrix.

Possible set of vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

For each matrix $A$ in Exercise 17 through 22, describe the image of the transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically (as a line, plane, etc. in ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$