mat223h-a10 - University of Toronto Department of...

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Unformatted text preview: University of Toronto Department of Mathematics MAT223H1Y—Summer 2010 Linear Algebra 1 Final Examination Thurs, August 19, 2010 J. Cho, E. Mazzeo Duration: 3 hours Last Name: Given Name: Student Number: Tutorial Section: No calculators or other aids are allowed. FOR MARKER USE ONLY Question Mark P 1 / 10 2 / 10 3 / 10 4 / 10 5 / 10 6 / 10 7 / 10 8 / 10 9 / 10 10 /10 TOTAL / 100 10f21 For each of the statements below, decide if it is true or false. Indicate your answer by shading in the box corresponding to your choice. Then justify your answer. [5] 1(3) S: {1+x,3a:+:z:2,2+x—$2} isabasis for PAR). false Justify your answer. 2of21 [5] 1(b) Let B be a fixed matrix in Mnxn(R). Then, S = {A E Mnxn(R)|AB = 0} is a subspace of Mnxn(R). Justify your answer. true false 30f21 [5] 2(a) Let W1, W2 be subspaces of a vector space V. Then their intersection W1 fl W2 is also a subspace of V. Justify your answer. true 4of21 [5] 2(b) Let V, W be vector spaces of dimension 2 and 3 respectively. Then there is a linear transformation L : V —> W which is onto. Justify your answer. 50f21 3 —2 1 O , and let b = 1 —1 3. Let A: r—tr—lw Nl—‘M Moor—- [4] (a) Solve the system An: = b and write your solution as a column vector. [2] (b) Find a basis for the null space of A. [2] (c) Find the rank of A. [2] (d) Which columns in A form a basis for the column space of A? 6of21 EXTRA PAGE FOR QUESTION 3. Please do not remove. 7of21 1 2 ——1 3 [6] 4(a) Let A = g L: g (1) . Compute the det(A). —l O —2 7 [4] 4(b) Let S = {(1,2,—1,3),(3,5,2,0),(0,1,3,1),(—1,0,—2,7)}. Determine whether 5' is linearly independent or not. Justify your answer. 8of21 EXTRA PAGE FOR QUESTION 4. Please do not remove. 90f21 1 —2 O 3 —4 . . 3 2 8 1 4 5. Let A be a given 4 X 4 matrlx. Let S — 2 , 3 , 7 , 2 , 3 —l 2 0 4 —4 Assume that every vector in S belongs to the null space of A. Then, [7] (a) Find a subset T of S which is a basis for SpanS. [3] (b) Determine whether the set of row vectors of A are linearly independent or not. 10 of 21 EXTRA PAGE FOR QUESTION 5. Please do not remove. 110f21 6. Let V = R4. Consider the inner product (u, v) = 4711121 + 3112122 + 2u3v3 + 11,4114 on V, Whereu=[u1 Mg 113 U4],V=[U1 ’02 v3 v4 LetW=spcm{X,y},wherex=[10 —1 0 ],y= [ —1 0 1 2 [7] (a) Find orthonormal bases for W and WL respectively. [3] (b) Find the orthogonal projection of z = [ 1 1 1 1 ] on W. 12 of 21 EXTRA PAGE FOR QUESTION 6. Please do not remove. 13 of 21 7. Let A: l—‘I—‘O r—Icw—A OHH [2] (3) Find the eigenvalues of A. [5} (b) Find a basis of eigenvectors for A. [3] (c) Find a matrix Q such that Q‘lAQ is diagonal. 14 of 21 EXTRA PAGE FOR QUESTION 7. Please do not remove. 15 of 21 8. Define T : nggflR) —+ BUR) by T (Z = (a + b) + (2d)a: + bx? Let [3 be the standard basis for M2x2<R>7 and let 7 be the standard basis for PAR). [4] (a) Prove that T is a linear transformation. [4] (b) Compute [2] (C) Find the null space of T. 16 of 21 EXTRA PAGE FOR QUESTION 8. Please do not remove. 17 of 21 [5] 9(a) Let V = MMHUR). For each ordered pair of matrices A, B in ManUR), we assign a real number (A, B) = T7"(BTA), where T7" denotes the trace of a matrix. Prove that (A, B) is an inner product on V. 18 of 21 [5] 9(b) Let V = R” be an inner product space with the standard inner product. Let A E MnanR). Assume that the rows of A form an orthonormal basis for IR”. Prove that AAT = In. 19 of 21 [10] 10. Let A be a square matrix of any size n X n. Suppose A is diagonalizable, and 1 and —1 are the only eigenvalues of A. Prove that A is invertible, and that A"1 = A. 20 of 21 EXTRA PAGE FOR QUESTION 10. Please do not remove. 21 of 21 ...
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This note was uploaded on 04/09/2012 for the course MATH 223 taught by Professor P during the Spring '11 term at University of Toronto- Toronto.

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mat223h-a10 - University of Toronto Department of...

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