Midterm 1-33A 2009 Fall

# Midterm 1-33A 2009 Fall - MATH 33A LECTURE 2 15T MIDTERM 2...

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Unformatted text preview: MATH 33A LECTURE 2 15T MIDTERM 2 Problem 1. (True/ False, 2 pts each) Mark your answers by filling in the appropriate box next to each question, 6 5 9 0 0 1 (iii If A iS an invertible n X 11 matrix, then the rank of A is 11. (iii: There exists a matrix A for which ker A is the same as the image of A. (iVI If AB 2 BA for two 11 x n matrices A, B, then either A or B must be the identity 2 3 7 1 O —1 (i! The reduced row echelon form of the matrix [1 4 8] is [0 1 O matrix. 1 O -—1 (V3 Asssume that the reduced row echelon form of A is O 1 O . Does the sys— 0 O O tem of linear equations A56 = 0 have more than one solution? (Vii If A - A - A is the identity matrix, then A is invertible. (vii: The columns of a matrix A always form a basis for the image of A. (Viﬁi If T and S are two linear transformations whose matrices [T] and [S] are equal, then T = 8. (ix: LEt A be an n x 111 matrix and B and m X 71 matrix (here it may be that n aé m)- If A is not invertible, is it true that also AB is not invertible? (x: The Set {(x,y) : x2 + y2 = 1} is a subspace of R2. 00/312 M t ‘ l” ‘ M1 -‘ J ° ‘ ‘f‘f‘ﬂm‘r’t‘ii on Q aw ﬂ. is W” l m i: t LO Tm: A (Bavariku is?) V m? 90 \mj‘jﬁz’ Yb w) Tm.- A: [59;] 42% 422M 1 Mrs} HUI) if) {7 ToJiis RQ=Q7QS ,: #5": it (VCQQTW'. T e e so 200%; *%,~:i 24' Q9 PM: D} to it = m s Cwuwtiﬂoh (A h r ti = (K) Faﬁali ba‘ (OHS in, tin tau} “i‘iﬁgii? "’5’ “5% a l I r ﬂ {3 e {o Q‘S MATH 33A LECTURE 2 15T MIDTERM 3 Problem 2. (20 pts) Let A = 1 3 —1 3 6 independent? (b) Find a basis for the image of A. (c) Find the kernel of A. 7 1 8 O . (a) Are the columns of A linearly 9 1 > i] veil.“ € (0‘) M01 :9} 2 Y3. 3,; r0 W9 Calumemxz 0% 0M3 L233 {933 N wag}? ‘ U ‘l \ \ g L} “z \ EB 2Q ’ \ 9:. 5 I]! a W O O \ \ —\ r“ 03 _. O Q m we 2 oo \ i ..\ ‘ " O \ O N x o 1. _ 6 O 3 Q 3 N o \ 0 o 1 l ~\ 0 o J "z ‘5‘! X1: Y5“:— ‘61 N,“ C? E52 A ,. "c \ e 1 "m ‘ X2?- ‘ K11,”ng " ’— JC‘ 2— ; i ’{Z\ \\ {O < 0 ‘ X32 -XLMYS. t; --¥:\“'JC§ ? 1L": ’ “4 MATH 33A LECTURE 2 ET MIDTERM 4 1 Problem 3. (20 pts) Let L be the line spanned by the vector 2 . (a) Find a matrix A so 3 that L is the image Of A- 0’) Find a matrix B so that L is the kernel of B. O \ o o t Phﬁ A : ’2- b m ; M5431 (3% A a ‘3 T“ K 3 0 <3 f. \ S90“ 0“; :‘VS (0"me5/ \/Q_ o‘\ P 3 ‘ 1 QXQAM L To (444.: VB}; (7“ \ W eivw’w: ><~ x K?) \A‘ - E - 33:1 Li“_.\.—Xl’:3/X1 ‘ 7w ‘ W "L? ILI H 1 In: ‘ X -S X?" g'lil‘é‘? : dﬁhti‘l @x3 Xg'h‘Eﬂ-qh’ —}_xl—}.x1+§:x IUI ’1 W N (L! 15 -1: “3. E : “1 g “4 1 MATH 33A LECTURE 2 IST MIDTERM 6 Problem 5- (20.1%) Let A be an n X 111 matrix, and let B be an invertible m X m matrix. (f1) Show that the Image of A is the same as the image of AB. (b) Express the rank of AB In terms of the rank of A and the rank of B. We“ 69"“ 33 ‘3 MM w W“ W X“ R (Mva \dm’\ 5: 3 f3 gnu [\7) r; tnwhkbt; Maia Q\SO \AW‘Q ARKE’\ :3, “bake guy“ \8 \S COMMSK\\'\ ‘S 7i 19 LA. 3AA.“ W 3) OX C) t“ M Am : 7km may; LR "M‘C‘qf a? ...
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