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Chapter 3: Q25E (page 120)

Describe the images and kernels of the transformations in Exercisesthrough geometrically.

25. Rotation through an angle蟺4 of in the counterclockwise direction (in鈩�2鈩�2).

Short Answer

Expert verified

The kernel is 0鈫�, image is all of 鈩�2.

Step by step solution

01

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If fis a function from X to Y, then

image(f)=fx:xinX=binY:b=f(x),forsomexinX

The Kernel of the transformation results in 0鈫�, as the reflection is its own inverse.

As the inverse of the reflection exists thus, the image will be 鈩�2鈩�2鈩�2. The kernel is of dimension 0鈫� so, the image will be of 鈩�2.

02

Final answer.

The kernel is 0鈫�, image is all of 鈩�2鈩�2鈩�2.

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Most popular questions from this chapter

Give an example of a parametrization of the ellipse

x2+y24=1

in鈩�2 . See Example .

We are told that a certain5脳5 matrixA can be written as

A=BC,

where Bis 5脳4andC is 4脳5. Explain how you know thatA is not invertible.

(a) Let Vbe a subset of role="math" localid="1660109056998" 鈩�n. Let mbe the largest number of linearly independent vectors we can find in V. (Note m鈮�n, by Theorem 3.2.8.) Choose linearly independent vectors 蠀鈫�1,鈥�鈥�蠀鈫�2,鈥�鈥�,鈥�鈥�蠀鈫�m inV. Show that the vectors 蠀鈫�1,鈥�鈥�蠀鈫�2,鈥�鈥�,鈥�鈥�蠀鈫�mspanV and are therefore a basis of V. This exercise shows that any subspace of鈩�n has a basis.

If you are puzzled, think first about the special case when role="math" localid="1660109086728" Vis a plane in 鈩�3. What ism in this case?

(b) Show that any subspaceV of 鈩�ncan be represented as the image of a matrix.

Find the basis of subspace of 鈩�4that consists of all vectors perpendicular to both

[10-11]and[0123] .

See definition A.8 in the Appendix.

Give an example of a matrixAsuch thatim(A)is spanned by the vector[15].
See all solutions

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