Short answer/true or false section. For the following problem, assume A is an n
ID: 3034659 • Letter: S
Question
Short answer/true or false section. For the following problem, assume A is an n times n matrix, B is an n times m matrix, x_i are n times 1 column vectors, and y_i are m times 1 column vectors. (a) Of the following operations, which make sense for arbitrary m, n? (a) AB, (b) BA, (c) (A + B)x_1, (d) x_1 middot x_2, (e) Ax_1 middot By_1, (f) x_2 middot y_2. (b) True/false. Two perpendicular vectors v_1, v_2 in R^2 form an independent set. (c) True/false. Two independent vectors v_1, v_2 in R_2 are also perpendicular. (d) What condition on the columns of A allows us to solve a system Ax = b by "dividing by A" in some sense? That is, when docs another matrix A^-1 exist satisfying x = A^-1 b? (c) Let u, v be unit vectors such that the angle between u and v is less than pi/2. What can I say about the size of u middot upsilon ? (f) A symmetric matrix is a matrix A such that A(i, j) = A(j, i) for all i, j. Write out a 3-by-3 symmetric matrix such that A > 0 (all entries are positive).Explanation / Answer
(a) AB is a nxm matrix, BA does not make sense (unless m=n), A+B and, therefore, (A+B)xi does not make sense as A and B are matrices of different sizes( A + B is not defined/does not exist), x1 .x2 does not make sense, Ax1 is a n x 1 matrix, By1 is a nx1 matrix so Ax1 . By1 does not make sense, x2 .y2 alsao does not make sense.
(b) Two perpendicular vectors in R2 make an independent set – TRUE
(c) The statement is FALSE ( the vectors (1,2)T and (3,4)T are independent, but not perpendicular).
(d) If is the angle between u and v, then cos = (u.v)/ IIu II IIvII or, u.v = cos a s u, v are unit vectors. Further since < /2, hence 0 < u.v (= co
(e) An example is as under:
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