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Chapter 3: Q35E (page 144)

Consider a non-zero vector 蠀鈫� in 鈩�n. What is the dimension of the space of all vectors in 鈩�nthat are perpendicular to 蠀鈫�?

Short Answer

Expert verified

The dimension of the space of all vectors in鈩�n is n鈭�1.

Step by step solution

01

To mention given data

Let x鈫�鈭�鈩�nbe a nonzero vector.

We need to find the dimension of the space of all vectors in鈩�n that are perpendicular to蠀鈫� .

02

To find the dimension of the space

Now, the vector x鈫�鈯�蠀鈫�.

Therefore,

蠀鈫�鈰�x鈫�=0鈬�蠀1x1+蠀2x2+鈰�+蠀nxn=0,

where,蠀i are the components of the vector 蠀鈫�.

Thus, by Exercise 33, these vectors form a hyperplane in 鈩�n.

Hence, the dimension of the space of all vectors in 鈩�nis n-1.

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Most popular questions from this chapter

Consider linearly independent vectors蠀鈫�1,蠀鈫�2,...,蠀鈫�p in a subspaceV of鈩�n and vectorsw1,w2,...,wq that span V. Show that there is a basis ofV that consists of all theu鈫�i and some of the wj鈫�. Hint: Find a basis of the image of the matrix

A=[||||v鈫�1...v鈫�1w鈫�1...w鈫�q||||]

How many cubics can you fit through nine distinct pointsP1(x1,y1),......,P9(x9,y9)?. Describe all possible scenarios, and give an example in each case.

Find a basis of the image of the matrix [012003000104000015000000].

In Exercises 37 through 42 , find a basisJ of 鈩�n such that theJ-matrix of the given linear transformation T is diagonal.

Orthogonal projection T onto the line in鈩�3 spanned by[123].

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

53.(0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(0,2),(1,2),(3,2).
See all solutions

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