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Chapter 3: Q37E (page 144)

Give an example of a4脳5matrix A withdim(ker(A))=3dim(ker(A))=3.

Short Answer

Expert verified

An example of 4脳5matrix suchA thatdim(ker(A))=3 is,

A=[10000010000000000000]

Step by step solution

01

To mention given data and recall theorem 3.3.7

Let Abe any4脳5 matrix.

We have to find the matrixA such that dim(ker(A))=3.

Theorem 3.3.7, is stated as follows:

For anyn脳m matrix A, the equation,

dim(ker(A))+dim(im(A))=m.

Here, in this case n=4,鈥夆赌�m=5.

02

To give an example of4×5 matrix with dim(ker(A))=3

By Theorem 3.3.7, we have,

dim(ker(A))+dim(im(A))=m鈬�dim(ker(A))+rank(A)=m鈬�dim(ker(A))=m鈭�rank(A)鈬�dim(ker(A))=5鈭�rank(A)

Consideringrank(A)=2 , whereA is any4脳5 matrix whose rank is 2 given by,

A=[10000010000000000000].

Hence, we get,

dim(ker(A))=5鈭�2鈬�dim(ker(A))=3

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Most popular questions from this chapter

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

18.1151012201230124

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.

50. (0,0),(1,0),(2,0),(3,0),(0,1),(0,2),(0,3),(1,1),(2,2),(3,3).

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||||]

Consider a nonzero vector 蠀鈫�in 鈩�3.Arguing geometrically, describe the image and the kernel of the linear transformation Tfrom 鈩�3to 鈩�to given by,

role="math" localid="1659526111480" T(x鈫�)=蠀鈫�路x鈫�.

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.

55.(0,0)(1,0)(2,0)(0,1)(1,1)(2,1)(0,2)(1,2)(2,2),(3,3).
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