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Chapter 6: Q57E (page 292)

Consider a linear transformation T from Rm+n to Rm . The matrix Aof T can be written in block form as A=[A1A2], where A1is m脳mand A2is m脳n. Suppose that det(A1)鈮0. Show that for every vector x鈫in Rn there exists a unique y鈫in Rmsuch that T[y鈫x鈫]=0鈫.Show that the transformationx鈫鈫y鈫from Rn to Rm is linear, and find its matrix M (in terms of A1and A2). (This is the linear version of the implicit function theorem of multivariable calculus.)

Short Answer

Expert verified

Therefore, the matrix M in terms of A1 and A2 is given by,A1M=-A2 .

Step by step solution

01

Matrix Definition 

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be a 鈥m by n鈥 matrix, written 鈥m脳n .鈥

02

To find M  in terms of A1 and A2 

Letx鈫鈭鈩n.

This corresponds to the last coordinates ofv鈫鈭鈩m+n.

So we can writev鈫=y鈫x鈫,

Where,

y鈫鈭鈩m.

We want to find such y鈫鈭鈩2that:

Tv鈫=A1A2y鈫x鈫

Tv鈫=0.

Clearly, we're looking for the unique y鈫such that A1y鈫=-A2x鈫.

This makes for a linear transformation from 鈩nto 鈩m, for whose matrix M鈭鈩m+napplies A1M=-A2.

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Most popular questions from this chapter

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

4.[1-12-2-1216211410-261033]

(For those who have studied multivariable calculus.) Let Tbe an invertible linear transformation from鈩2to鈩2, represented by the matrix M. Let惟1be the unit square in 鈩2and惟2its image under T . Consider a continuous functionf(x,y)from鈩2to鈩, and define the functiong(u,v)=f(T(u,v)). What is the relationship between the following two double integrals?

鈭惟2f(x,y)dAand鈭惟1g(u,v)dA

Your answer will involve the matrix M. Hint: What happens whenf(x,y)=1, for allx,y?

Find the determinants of the linear transformations in Exercises 17 through 28.

21.T(f(t))=f(-t)fromP3toP3

Find the determinants of the linear transformations in Exercises 17 through 28.

24.T(z)=(2+3i)zfrom鈩to鈩

What are the lengths of the semi axes of the largest ellipse you can inscribe into a triangle with sides 3,4 , and 5 ? See Exercise 48.
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