Linear Algebra : Matrices

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #33 : Vector Vector Product

Which of the following applies to  ,  where "  " and "" refer to the dot product and the cross product of two vectors?

Possible Answers:

 

 is an undefined expression.

Correct answer:

 

Explanation:

The cross product of  two vectors in  is also a vector in . It follows that  and ; it further follows that.

Example Question #34 : Vector Vector Product

Which of the following applies to  ,  where "" refers to the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

 is an undefined expression.

Correct answer:

 is an undefined expression.

Explanation:

The cross product of two vectors is defined only if both vectors are in  and  are vectors in , so  is undefined; consequently, so is .

Example Question #35 : Vector Vector Product

Which of the following applies to  ,  where "  " refers to the dot product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

 is an undefined expression.

Correct answer:

Explanation:

The dot product of two vectors in the same vector space is a scalar quantity. , so   and  are in the same vector space; their dot product is defined, and . For similar reasons, . Therefore, their sum is defined, and .

Example Question #35 : Vector Vector Product

.

Which of the following applies to ,  where "  " and "" refer to the dot product and the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

 is an undefined expression.

Correct answer:

 is an undefined expression.

Explanation:

The cross product  of two vectors in  is also a vector in . The dot product  of two such vectors is a scalar. Since a vector and a scalar cannot be added,  is an undefined expression.

Example Question #161 : Matrices

True or false: It follows that .

Possible Answers:

True

False

Correct answer:

True

Explanation:

One property of vector dot products is commutativity - that is, 

.

Therefore, if , then .

The statement is true,

Example Question #162 : Matrices

True or false: It follows that .

Possible Answers:

False

True

Correct answer:

False

Explanation:

One property of vector cross products is anticommutativity - that is, 

.

If , it follows that  

.

The statement is false.

Example Question #38 : Vector Vector Product

Which of the following applies to  ,  where "  " and "" refer to the dot product and the cross product of two vectors?

Possible Answers:

 is an undefined expression.

Correct answer:

Explanation:

The cross product of two vectors in  is also a vector in . It follows that  and . The dot product of two vectors in the same vector space, is a scalar, so , the  dot product of two vectors in , is a scalar in .

Example Question #40 : Vector Vector Product

Find  and  so that .

Possible Answers:

There does not exist any such  and 

Correct answer:

Explanation:

The cross product of two vectors in  can be set up and calculated as if it were a determinant of a matrix with its top row comprising , the unit vectors of , and the other two comprising the elements of  and :

Calculate as you would a determinant, adding the upper-left to lower-right products and subtracting upper-right to lower-left products: 

Cross product

The cross-product is equal to 

We want this vector to be equal to , so the following must hold:

 

 

Examining the third equation, , we find that this is consistent with the other equations, since  and  make this true. Therefore,  and  are the values sought. 

Example Question #41 : Vector Vector Product

.

Which of the following holds for  ?

Note:  (boldface) refers to the vector 

Possible Answers:

, but the result need not be 

, but the result need not be 0.

 is an undefined expression.

, where 0 refers to the real zero.

, where   refers to the vector .

Correct answer:

, where   refers to the vector .

Explanation:

One property of vector cross products is anticommutativity - that is, 

.

It therefore follows that for all 

, the zero vector .

Example Question #42 : Vector Vector Product

A parallelogram has these two vectors as sides. Find  so that the parallelogram is a rhombus. 

Possible Answers:

Correct answer:

Explanation:

For the parallelogram formed by  and  to be a rhombus, the vectors must be  of equal length, or norm - . The norm of a vector is equal to the square root of the sum of the squares of its entries; to set the norms equal, it suffices to set the squares of the norms equal, and to solve for :

 

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