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Chapter 3: Q31E (page 120)

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

Short Answer

Expert verified

-2-30110 is a matrix A such thatimA is the plane with normal vector132 .

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 f is a function from X to Y, then

imagef=fx:xinX=binY:b=f,forsomexinX

Consider the system in terms of an equation,x+3y+2z=0鈬x=-3y-2z

When, y=0鈬x=-201

When, z=0鈬x=-310

Thus, -201,-310are the required vectors.

02

Final answer.

-2-30110 is a matrix A such that im(A) is the plane with normal vector132 .

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Most popular questions from this chapter

Express the image of the matrix

A=[1116123413521470]

as the kernel of a matrixB. Hint: The image ofAconsists of all vectorsy鈫in鈩4such that the systemAx鈫=y鈫is consistent. Write this system more explicitly:

localid="1664197199135" |x1+鈥鈥鈥鈥x2+鈥鈥鈥鈥x3+6x4=y1x1+2x2+3x3+4x4=鈥y2x1+3x2+5x3+2x4=鈥y3x1+4x2+7x3鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥=鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥y4|.

Now, reduce rows:

x1-鈥鈥鈥鈥鈥鈥鈥鈥x3+8x4=4y3-3y4x2+2x3-2x4=鈥鈥鈥鈥鈥-y3+y40=鈥鈥鈥鈥鈥-3y3+2y4鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥0=鈥鈥y2-鈥2y3+y4

For which the vectors y鈫is this system consistent? The answer allows you to express im ( A) as the kernel of a2脳4matrix B.

Given a hexagonal tiling of the plane,such as you might find on a kitchen floor, consider the basisBof鈩2consisting of the vectorsv鈫,w鈫in the following sketch:

(a) Find the coordinate vectors[OP炉]aand[OQ鈫]a.Hint: Sketch the coordinate grid defined by the basisB=(V鈫,w鈫).

(b) We are told that[OR鈫]B=[32]. Sketch the pointR. IsRa vector or a center of a tile?

(c) We are told that[OS鈫]B=[1713]. IsSa center or a vertex of a tile?

If A and B are invertible matrices, then AB must be similar to BA.

Express the plane V in 鈩3with equation 3x1+4x2+5x3=0as the kernel of a matrixA and as the image of a matrix B.

If the kernel of a matrix A consists of zero vector only, then the column vectors of A must be linearly independent.
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