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Chapter 1: Q3.4-18E (page 1)

In Exercises1 through 18, determine whether the vector x鈫� is in the span V of the vectors v1鈬赌,,,,vm鈬赌(proceed 鈥渂y inspection鈥� if possible, and use the reduced row-echelon form if necessary). If x鈫�is in V, find the coordinates of x鈫�with respect to the basis role="math" localid="1664193712120" I=v1鈬赌,,,,vm鈬赌of V, and write the coordinate vector [x鈬赌]I.

role="math" localid="1664193998673" x鈬赌=[5432];v1鈬赌=[1101];v2鈬赌=[0110];v3鈬赌=[0-101]

Short Answer

Expert verified

As the coordinates of the vector are not available, so, x鈬赌I isn鈥檛 in V.

Step by step solution

01

Consider the vectors.

The vectors are,

x鈬赌=[5432];v1鈬赌=[1101];v2鈬赌=[0110];v3鈬赌=[0-101]

02

Check for the coordinates of the vector.

The formula is, x鈬赌=c1v1鈬赌+c2v2鈬赌.

Consider the matrix form and perform rrref,

rref100511-1401030012=1000010000100001

03

Final answer.

As the coordinates of the vector are not available, so, x鈬赌I isn鈥檛 in V .

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Most popular questions from this chapter

For an arbitrary positive integern鈮�3, find all solutions x1,x2,x3,......,xnof the simultaneous equations x2=12(x1+x3),x3=12(x2+x4),.....,xn鈭�1=12(xn鈭�2+xn). Note that we are asked to solve the simultaneous equations xk=12(xk鈭�1+xk+1), fork=2,3,.....,n鈭�1 .

Consider a linear system of four equations with three unknowns. We are told that the system has a unique solution. What does the reduced row-echelon form of the coefficient matrix of this system look like? Explain your answer.

Make me a crown weighing 60 minae from a mixture of gold, bronze, tin, and wrought iron. Let the gold and bronze together form two-thirds of the weight, the gold and tin together three-fourths, and the gold and iron three-fifths. Tell me how much gold, tin, bronze, and iron you must use. (From the Greek Anthology by Metrodorus, 6thcentury A.D.)

Balancing a chemical reaction. Consider the chemical reaction

aNO2+bH2O鈫�cHNO2+dHNO3,

where a, b, c, and d are unknown positive integers. The reaction must be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same,

2a+b=2c+3d.

While there are many possible values for a,b,c and d that balance the reaction, it is customary to use the smallest possible positive integers. Balance this reaction.

In Exercises1 through 18, determine whether the vector x鈫� is in the span V of the vectors v1鈫�,,,,,vm鈫�(proceed 鈥渂y inspection鈥� if possible, and use the reduced row-echelon form if necessary). If localid="1664192105355" x鈬赌is in V, find the coordinates of x鈬赌with respect to the basis I=v1鈬赌,....vm鈬赌of V, and write the coordinate vector [x鈫�]I.

localid="1664193284963" x鈫�=[111-1];v1鈫�=[1020];v2鈫�=[0130];v3鈫�=[0041]
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