07spr-m1sols

# 07spr-m1sols - Math 51 Spring 2007 Exam 1 Solutions Page 1...

This preview shows pages 1–4. Sign up to view the full content.

Math 51, Spring 2007 Exam 1 Solutions — April 24, 2007 Page 1 of 9 1. (10 points) Complete each of the following sentences. (a) A collection of vectors -→ v 1 , . . . , -→ v k is defined to be linearly independent if (5 points) . . . the equation c 1 -→ v 1 + c 2 -→ v 2 + · · · + c k -→ v k = -→ 0 for scalars c 1 , . . . , c k implies c 1 = · · · = c k = 0. OR: . . . no vector in the collection can be written as a linear combination of the other vectors. (b) A basis for a subspace V is defined to be (5 points) . . . a linearly independent set (or collection) of vectors whose span is V . (Note: saying only the portion “linearly independent set” earned 3 points; saying only the portion “set that spans V ” also earned 3 points.) OR: . . . a collection C of vectors such that any vector in V can be uniquely expressed as a linear combination of vectors in C .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Math 51, Spring 2007 Exam 1 Solutions — April 24, 2007 Page 2 of 9 2. (10 points) Let Q be the set of all vectors in R 3 that are orthogonal to -→ w = (1 , 3 , - 1). (a) Find a parametrization for the set Q . (Hint: Q forms a plane in R 3 .) (5 points) If -→ v = ( x, y, z ) is orthogonal to -→ w , then -→ v · -→ w = 0, i.e., x + 3 y - z = 0. This is a “system” with two free variables y and z , so that x y z = - 3 y + z y z = - 3 y y 0 + z 0 z = y - 3 1 0 + z 1 0 1 . Equivalently, Q = s - 3 1 0 + t 1 0 1 s, t R = x y z x = - 3 s + t, y = s, z = t . (b) Suppose P is a plane in R 3 , parallel to Q , such that P passes through the point (2 , 0 , - 1). Find an equation for P , written in the form ax + by + cz = d . (5 points) Solution 1: The plane Q has normal vector -→ w , since every vector in Q is orthogonal to -→ w . Since P is parallel to Q , it also has normal vector -→ w . Now if ( x, y, z ) is any point in P , then since (2 , 0 , - 1) also lies in P , we know that the vector -→ w is orthogonal to the vector -→ u = x - 2 y z + 1 from ( x, y, z ) to (2 , 0 , - 1). Thus, 0 = -→ w · -→ u = ( x - 2) + 3 y - ( z + 1) = x + 3 y - z - 3 , so that x + 3 y - z = 3 is the equation for P . (Note: In this instance, we were given -→ w as a normal vector to Q , but in other situations we might need to construct such a vector using Q ’s parametrization. In such a case, remember this can be done by taking the cross product of any two non-collinear vectors that span Q .) Solution 2: Since P is parallel to Q , the plane P in parametric form is 2 0 - 1 + s - 3 1 0 + t 1 0 1 s, t R , or equivalently x y z x = 2 - 3 s + t, y = s, z = - 1 + t . We can combine the above equations to eliminate s and t , and we find x = 2 - 3 y + ( z + 1) , which simplifies to x + 3 y - z = 3 , as before.
Math 51, Spring 2007 Exam 1 Solutions — April 24, 2007 Page 3 of 9 3. (10 points) Compute, showing all steps, the reduced row echelon form of the matrix 1 0 2 1 0 - 1 0 - 2 0 - 1 2 2 2 3 7 - 2 2 - 6 0 6 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern