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Chapter 3: Q3.1-40E (page 120)

Consider an 鈥�n脳pmatrix鈥�Aand p脳mmatrix B. If ker (鈥�A) = im (B), what can you say about the product AB?

Short Answer

Expert verified

The product AB=0.

Step by step solution

01

To define kernel of linear transformation

Thekernel of linear transformationis defined as follows:

The kernel of a linear transformationTx鈫�=Ax鈫� from鈩�m to鈩�n consists of all zeros of the transformation, i.e., the solutions of the equations Tx鈫�=Ax鈫�=0.

It is denoted by kerT or kerA.

LetA be ann脳p matrix andB be anp脳m matrix.

02

Determine about the product 

Let x鈫�鈭�鈩�m.

Therefore, Bx鈫�鈭�imB=kerA.

This implies that

ABx鈫�=ABx鈫�=0.

Take x鈫�=e鈫�i鈭�鈩�m.

Thus, we get,

ABe鈫�i=0鈫�,鈥�鈥�鈥�i=1,鈥�2,鈥�鈥�,鈥�m.

This implies that all columns of ABare 0.

Hence, the product AB=0.

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Most popular questions from this chapter

For which value(s) of the constant k do the vectors below form a basis of 鈩�4?

[1002],[0103],[0014],[234K]

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.

54.(0,0)(1,0)(2,0)(0,1)(1,1)(2,1)(0,2)(1,2)(2,2).

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45. Show that the cubic through the points(0,0),(1,0),(2,0),(3,0),(0,1),(0,2)补苍诲鈥�(0,3) can be described by equations of the form c5xy+c8x2y+c9xy2=0, where at least one of the coefficientsc5,c8,补苍诲鈥�c9 is nonzero. Alternatively, this equation can be written asxy(c5+c8x+c9y)=0 . Describe these cubic geometrically.

Give an example of a matrixAsuch thatim(A)is the plane with normal vector [132]in鈩�3鈩�3 .

Question: Consider three linearly independent vectorsv1鈫�,v2鈫�,v3鈫�in 鈩�3. Are the vectorsv1鈫�,v1鈫�+v2鈫�,v1鈫�+v2鈫�+v3鈫�linearly independent as well? How can you tell?
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