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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q36E (page 73)

Let Tbe an invertible linear transformation fromR² to R². Let Pbe a parallelogram inR². Is the image of Pa parallelogram as well? Explain.

Short Answer

Expert verified

Yes, the image of P is also a parallelogram as well.

Step by step solution

01

Consider the linear transformation

The vector form of the parallelogram is,

$\overset{鈫�}{p} = a \overset{鈫�}{p_{1}} + b \overset{鈫�}{p_{2}}$

The linear transformation is,

$T (\overset{鈫�}{p}) = T (a \overset{鈫�}{p_{1}} + b \overset{鈫�}{p_{2}})$

02

Compute the condition

Consider the linear transformation is,

$T (\overset{鈫�}{p}) = T (a \overset{鈫�}{p_{1}} + b \overset{鈫�}{p_{2}})$

Use the properties of the linear transformation in the above equation.

$\begin{array}{l} T (\overset{鈫�}{p}) = T (a \overset{鈫�}{p_{1}}) + T (b \overset{鈫�}{p_{2}}) \\ 鈬� T (\overset{鈫�}{p}) = a (T \overset{鈫�}{p_{1}}) + b (T \overset{鈫�}{p_{2}}) \end{array}$

This represents that the image of P is also a parallelogram as well.

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Most popular questions from this chapter

TRUE OR FALSE?

There exists a matrix A such that $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] A [\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ .

Question 8: Find the inverse of linear transformation

$y_{1} = x_{1} + 7 x_{2} y_{2} = 3 x_{1} + 20 x_{2}$

if $A^{2} = I_{2}$ then matrix A must be either $I_{2} o r - I_{2}$ .

TRUE OR FALSE?

There exists an invertible matrix S such that $S^{- 1} (\begin{matrix} 0 & - 1 \\ 0 & 0 \end{matrix})$ is a diagonal matrix.

Use the formula derived in Exercise $2.1.13$ to find the inverse of the rotation matrix

localid="1659346816315" $A = [\begin{matrix} 肠辞蝉胃 & - 蝉颈苍胃 \\ 蝉颈苍胃 & 肠辞蝉胃 \end{matrix}]$ .

Interpret the linear transformation defined by $A^{- 1}$ geometrically. Explain.

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