Math 33A Winter 2008 Midterm I - UCLA Math 33A Lec 2 Winter...

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Unformatted text preview: UCLA Math 33A Lec 2, Winter 2008 Midterm 1 Feb 1, 2008 Last name: First and middles Names: g g w _ UCLA ID number: Section (circle one): 2A WONG, WANSHUN Tuesday 23 WONG, WANSHUN Thursday 20 KWON, SOONSIK Tuesday 213 KWON, SOONSIK Thursday Instructions: 0 FEB in your name ané circle your section above. a You are allowed to use your brain, pencii/ pen, eraser oniy. - You can use the blank pages if you need space to finish your solution. Please indicate cleariy Where it may be found. a Show ah the necessary steps involved in finding you: solutions1 except True~False questions. Problem 1: (10 pts) Indicate Whether each statement is true or false (circle your answer, and you need not justify your answer here). 1. 10. . There exists a 2 X 2 matrix A such that A m and A m . T 2 “ w 2 _ The function T m Gym? 2) (y 2) > is a linear transformation. K F . The matrix ( 1 3 0 1 ‘ Matrix 0 0 1 2 is in rref. F 0 O O O . If 11'} 'LU ere nonzero vectors in R3, then iii must be a linear combination of if and 17. T {F . if A 4 X then there exists a vector 3 E R4 such that the system A5 = I; is inconsistent. T F (A) < : (Ali—i), then the system of linear equations Af = Ernest be inconsistent. . If A is e 3 x 4 matrix of rank 3, then the system A55 = 2 1 must have infiniteiy many 3 solutions. (SE W4)? —— (m~i~4)2 IfAendBaretwoéiXBm 'es such that A6 = B??? for all vectors 1? E R3, then matrices A and B must be equal. " _6 5) represts a rotation combined with a scaling. The formula det(2A) m 2 det(A) holds for ail 2 X 2 matrix A. T ) ‘ I. (6 pés) Find the inverse of A. 221 010 230 Probiem 2: Let A = ( Problem 3: 1. (6 pts) Find a nonzero 2 x 2 matrix A such that Af is peraiiei to the vector (1) , for ali 1i? 6 R2. Find the inverse if exists. 2. (7 pts) Find the 3 x 3 matrix of the reflection in R3 about the inane y w 2. Find the inverse if exists. 3. (7 pts) Find the 2 X 2 matrix of the iinear transformation that rotates ail vector E E R2 by 120° in the countezciockwise direction, then dilates the resulting vector by a factor of Find the inverse if exists. Problem 4: (8 pts) Does there exist an invertible matrix that has a column of all zeros? If so then find one, if not then expiain why not. wiil be given for right answer but wrong reason.) ...
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