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Chapter 3: Q29E (page 143)

Find a basis of the subspace of 鈩3defined by the equation2x1+3x2+x3=0.

Short Answer

Expert verified

The basis of the subspace V of 鈩3defined by the equation 2x1+3x2+x3=0is 鈭320,鈭102.

Step by step solution

01

Finding the basis of 2x1+3x2+x3=0

Let V be a subspace鈩3 such that

V={(x1,x2,x3)鈭鈩3:2x1+3x2+x3=0}

鈬V={(2x1,2x2,2x3)鈭鈩3:2x1+3x2+x3=0}

鈬V={(鈭3x2鈭x3,2x2,2x3)鈭鈩3:2x1=鈭3x2鈭x3}

鈬V=x2鈭320+x3鈭102

鈬Basis of V=鈭320,鈭102

02

Final Answer

The basis of the subspace of 鈩3defined by the equation 2x1+3x2+x3=0is

鈭320,鈭102

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Most popular questions from this chapter

Consider the plane x1+2x2+x3=0. Find a basis of this plane such that [x鈫]B=[2-1]forx鈫=[1-11].

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?

The column vectors of a 5脳4 matrix must be linearly dependent.

Find a basis of the image of the matrix [012003000104000015000000].
See all solutions

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