midterm 1 spring 2003 solutions

Linear Algebra with Applications

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 21b Midterm I—Solutions Thursday, March 6, 2003 1. (12 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If { v 1 , v 2 , v 3 } is a linearly independent set in R n , then { v 1 + v 2 , v 2 + v 3 , v 1 + v 3 } is also a linearly independent set in R n . Solution. This statement is true. Suppose that a ( v 1 + v 2 )+ b ( v 2 + v 3 )+ c ( v 1 + v 3 ) = ( a + c ) v 1 +( a + b ) v 2 +( b + c ) v 3 = 0 . Then a + c = 0 a + b = 0 b + c = 0 , since { v 1 , v 2 , v 3 } is a linearly independent set. If we solve this system, then we see that a = b = c = 0. Therefore, the vectors v 1 + v 2 , v 2 + v 3 , v 1 + v 3 must be linearly independent. T F (b) It is possible to have a 5 × 3 matrix A such that the dimension of the kernel of A is four. Solution. This statement is false. Since rank( A ) + nullity( A ) = 3, the dimension of the kernel of A must be less than or equal to 3. T F (c) If A 2 + 2 A - 5 I 3 = 0 for a 3 × 3 matrix A , then A is invertible. Solution. This statement is true. Solving for I 3 , I 3 = A 1 5 A + 2 5 I 3 . Therefore, A - 1 = 1 5 A + 2 5 I 3 . T F (d) If A and B are n × n matrices and x is in the kernel of A , then x must also be in the kernel of AB . 1
Image of page 1

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

View Full Document Right Arrow Icon
Solution. This statement is false. Let A = 1 0 0 0 and B = 0 1 1 0 . Then x = 0 1 is in the kernel of A but is not in the kernel of AB . T F (e) Row operations on an m × n matrix A can change the kernel of A . Solution. This statement is false, since it statement contradicts Gauss-Jordan elimination. T F (f) If A and B are m × n matrices, then rank( A + B ) = rank( A ) + rank( B ) . Solution. This statement is false. Let A = 1 0 0 1 and B = - 1 0 0 - 1 . 2. (a) (5 points) Let A = 1 - 9 3 - 1 - 4 2 2 - 5 1 and y = 5 - 8 α For what value(s) of α , if any, will y be in the image of A .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern