This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MATH. 20F SAMPLE MIDTERM 1 SOLUTIONS You have 50 minutes for this exam. Please write legibly and show all working. No calculators are allowed. Write your name and ID number. Name: ID Number: (1) Let A = 1 2 1 1 a 2 1 3 . (i) (8 points) Find all a s such that the homogeneous system Ax = 0 has nontrivial solutions. After performing Gaussian elimination on A , one obtains the row echelon form 1 2 1 0 3 1 0 0 a 5 . It follows that there is a free variable if and only if a = 5. Hence, Ax = 0 has infinitely many solutions if and only if a = 5. (ii) (7 points) For those values of a from (i), find the general solution of Ax = 0. Taking a = 5 in the row echelon form from (i), we see that x 3 is a free variable. Setting x 3 = , we see that x 2 = 1 3 and x 1 = 5 3 . So the general solution is x = ( 5 3 , 1 3 , 1) t . 1 2 MATH. 20F SAMPLE MIDTERM 1 SOLUTIONS (2i) (5 points) Let T : R n R m be a linear transformation. If x and...
View Full
Document
 Winter '03
 BUSS
 Math, Linear Algebra, Algebra

Click to edit the document details