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Chapter 1: Q48E (page 40)

Question: Determine whether the statements that follow are true or false, and justify your answer.

48.If vectorw鈬赌 is a linear combination of u鈬赌andv鈬赌 , then u鈬赌+v鈬赌+w鈬赌 must be a linear combination ofu鈬赌 andu鈬赌+v鈬赌 .

Short Answer

Expert verified

The statement, 鈥淚f vector w鈬赌 is a linear combination of u鈬赌andv鈬赌 , then localid="1659960886081" u鈬赌+v鈬赌+w鈬赌 must be a linear combination of u鈬赌and u鈬赌+v鈬赌.鈥� is false.

Step by step solution

01

Consider the condition

Let the vectorw鈬赌 with linear combination of u鈬赌andv鈬赌 be, u鈬赌+v鈬赌+w鈬赌, where and are scalar.

Then, the vector, u鈬赌+v鈬赌+w鈬赌becomes,

u鈬赌+v鈬赌+w鈬赌=u鈬赌+v鈬赌+xu鈬赌+yv鈬赌=u鈬赌(1+x)+v鈬赌(1+y)

Then, the vector, u鈬赌+v鈬赌+w鈬赌 is a linear combination of u鈬赌andv鈬赌.

02

Conclusion

Hence, the vector u鈬赌+v鈬赌+w鈬赌 is a linear combination of u鈬赌andv鈬赌 and not u鈬赌 and u鈬赌+v鈬赌.

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Most popular questions from this chapter

Consider some particles in the plane with position vectors r1鈫�,r2鈫�,.....,rn鈫�and massesm1,m2,.....,mn .

The position vector of the center of mass of this system is

rcm鈫�=1M(m1r1鈫�+m2r2鈫�+.....+mnrn鈫�)

whereM=m1+m2+.....+mn .

Consider the triangular plate shown in the accompanying sketch. How must a total mass of be distributed among the three vertices of the plate so that the plate can be supported at the point[22] ; that is,rcm鈫�=[22] ? Assume that the mass of the plate itself is negligible.

In exercises 1 through 10, find all solutions of the linear systems using elimination. Then check your solutions

7. x+2y+3z=1x+3y+4z=3x+4y+5z=4

Let A be a 4 脳 3 matrix, and letb鈫�and c鈫� be two vectors in 鈩�4. We are told that the systemrole="math" localid="1659341825668" Ax鈫�=b鈫� has a unique solution. What can you say about the number of solutions of the systemAx鈫�=c鈫� ?

Compute the products Axin Exercises 13 through 15 using

paper and pencil. In each case, compute the product two ways: in terms of the columns of A and in terms of the rows of A.

13.

[1234][711]

Recall that a real square matrix A is called skew symmetric ifAT=-A.

a. If A is skew symmetric, isA2skew symmetric as well? Or isA2symmetric?

b. If is skew symmetric, what can you say about the definiteness of A2? What about the eigenvalues of A2?

c. What can you say about the complex eigenvalues of a skew-symmetric matrix? Which skew-symmetric matrices are diagonalizable over 鈩�?
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