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MATH 2510C Final 0708

# MATH 2510C Final 0708 - %”3(it 3 E Pagelof 3...

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Unformatted text preview: %”3 (it 3 E) Pagelof 3 : : llﬁl‘EFf/ﬁ IVs-mu 7?; ﬁg El] 1 j: ’3’ Co'pyrright Reéglxcg The Chinese University of Hong Kong LO 0: i :0 0/; \$5? _ eﬁsrilaeeea Course Examination M Term, 2001~ - 2008_ ﬁt 13 iii/3’42; 713:: . . Course Code & We: MAT2310C LINEAR ALGEBRA AND APPLICATIONS 3?} Fa'l Il‘ 53? ’IM‘E‘ Time allowed : ______________ _______________ 0 hours __________ ____________ __ minutes 5% i if; E»: 51% Student ID. No. : Seat N0. : Answer all the questions. 1. Let 2 —1 —1 1 2 A: —2 1 1 0 mg —4 4 10 8 0 —6 3 3 1 —4 (6 points) What is the rank of A and why? (b) (6 points) Find the null space of A? What is the nullity of A? (c) (6 points) What is the column space of A and Why? (d) (2 points) Find a condition on oz, ﬂ, 7, 7/ that is necessary and sufficient for 05 Ax : ﬂ 7 77 to have a solution. 1 1 2 1 _2 l 2.LetA: 0 and b: 4 ~1 1 —1 1 1 1 ~1 0 g (a) (8 points) Use the Gram—Schmidt process to ﬁnd an orthonormal basis for the column space of A. (b) (5 points) Find a QR—factorization of A. (c) (2 points) Use 2(b), solve the least squares problem Ax = b. MAT2310C » Course Code ﬂ 8 \$131? 5 ﬁ‘i 2 E (éé 3 E) Page 2 of 3 3. Let 5 —7 7 A: 4 —3 4 4 —1 2 (a) (10 points) Find all the eigenvalues of A and the corresponding eigenspace. (b) (8 points) Is A diagonalizable? Explain and if the answer is yes, ﬁnd a matrix P that diagonalizes A, and compute P‘1 and P“1AP. Then, (2 points) calculate 1 the matrix J = 5A4 — (A‘1)3 —1- 5L 4. Recall that P3 is the vector space of all polynomials p(t) of degree at most 3. Let L: A122 —> P3 be the linear transformation deﬁned by L( (3 points) Find the set of all vectors in ker L. ab cd D =(a—b+56+d)t3+(3a+2b+3d)t2 +(a+b~c+d)t+(2a+3b~50+2d). (b) (3 points) Find a basis for ker L. (c) (3 points) Is L is one—to-one? Explain. (d) (3 points) Is L onto? Explain. (e) points) Is L is invertible? Explain. 5. (10 points) Let W be the subspace of R4 that consists of all vectors of the form 27‘ ~s 47" — 28 2s — 27" T Find the vector in W that is Closest to 1 —1 1 —1 ] and the vector in WL that is closest to this same vector. . CourseCode gamma: .... .................................................... .. a 3 E (a 3 a) Page 30f 3 6. Let L : R3 —> R3 be deﬁned by x x—2y+4z L 2 2x+y z y—z Let S : {e1,e2,e3} be the natural basis for R3 and let 1 0 1 T: O , 2 , O 4 —1 2 be another basis for R3. (a) (5 points) Compute the matrix of L with respect to S and T. (b) (5 points) Compute the matrix of L with respect to T via the transition matrix. 7. Are the following statements true or false? You will not receive any mark unless you justify your answers. (2 points) If E is an elementary matrix, then det E 2 il. (b) (2 points) If there exists a linearly dependent set {V1,V2, - -- ,vp} in a vector space V, then dimV g p — 1. (c) (2 points) Let z be a vector in R" of length 1. The matrix Q 2 I — 2zzT is symmetric and orthogonal. (d) (2 points) Let K be a nonzero subspaces of a finite—dimensional vector space V and let L : V —+ W be a linear transformation of V to the vector space W. Then dim L(K) S dim K. (e) (2 points) If F and G are row equivalent square matrices, then the eigenvalues of F are the same as the eigenvalues of G. N THE END N ...
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MATH 2510C Final 0708 - %”3(it 3 E Pagelof 3...

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