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Chapter 3: Q30E (page 143)

Find a basis of the subspace of 鈩�4defined by the equation

2x1-x2+2x3+4x4=0.

Short Answer

Expert verified

The basis of the subspace V 鈩�4of defined by the equation2x1-x2+2x3+4x4=0is1200,0210,0401.

Step by step solution

01

Finding the basis

2x1-x2+2x3+4x4=0

Let V be a subspace 鈩�4such that

V=x1,x2,x3,x4鈭�鈩�4:2x1-x2+2x3+4x4=0鈬�V=x1,2x1+2x3+4x4,x3,x4鈭�鈩�4:x2=2x1+2x3+4x4

鈬�V=x11200+x30210+x40401

鈬� Basis of V=role="math" localid="1659413559674" V=1200,0210,0401

02

Final Answer

The basis of the subspace of defined by the equation 2x1-x2+2x3+4x4=0is1200,0210,0401.

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Most popular questions from this chapter

An n 脳 n matrix A is called nilpotent ifAm=0for some positive integer m. Examples are triangular matrices whose entries on the diagonal are all 0. Consider a nilpotent n 脳 n matrix A, and choose the smallest number 鈥榤鈥� such that Am=0. Pick a vector v鈫�in 鈩�nsuch that Am-1v鈫�鈮�0. Show that the vectorsv鈫�,Av鈫�,A2v鈫�,...,Am-1v鈫�are linearly independent.

Hint: Consider a relation c0v鈫�+c1Av鈫�+c2A2v鈫�+...+cm-1Am-1v鈫�=0. Multiply both sides of the equation with Am-1to show c0=0. Next, show thatc1=0,and so on.

Express the line Lin 鈩�3 spanned by the vector[111]as the image of a matrix Aand as the kernel of a matrix B.

Question: Consider linearly independent vectors v1鈫�,v2鈫�,....,vm鈫�in 鈩�nand let A be an invertible m脳mmatrix. Are the columns of the following matrix linearly independent?

[IV1IV2...IVmIII]A

Consider a linear transformation T from鈩�nto鈩�pand some linearly independent vectorsv鈫�1,v鈫�2,...v鈫�min鈩�n. Are the vectorsT(v鈫�1),T(v鈫�2),...T(v鈫�m)necessarily linearly independent? How can 鈩�pyou tell?

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

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