Linear Algebra with Applications

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Unformatted text preview: Last Name: First Name: Mathematics 21b First Midterm Examination Your Section (circle one): No calculators are allowed. October 25, 1999 John Boller MWF 10 Noam Elkies MWF 11 score - I | | I I - - - - -~—-——.—.-.—.._...___—_—._ - 1. Consider the system of linear equations 3x3 — 2mg = —3, $3 + 2234 + 42:1 : 1, 2331 +$2 ‘—SL'3 (a) (2 pts.) Write the coefficient matrix and augmented matrix for this system. (b) (5 pts.) Calculate the row-reduced echelon form of the augmented matrix. (C) (3 pts.) Find the general solution of the linear system. Verify that your answer does in fact satisfy all the equations. 2. Let A and B be the matrices 1 A20 0 coo I—LCDD col—a l—‘CDC car—1c; , B: . (a) (4 pts.) Describe A, B geometrically as linear transformations. (3 pts.) What are the ranks of A and B? Is either A or B invertible? Justify your answers. (0) (3 pts.) Do A and B commute? Interpret your result geometrical-1y. (a) (1 pt. each) Define: o kernel 0 image 7 0 rank 0 span 0 basis (b) If A is a 4 X 5 matrix of rank 4,. o (1 pt.) Is A invertible? Why? 0 (1 pt.) What can you say about ker(A)? o (1 pt.) What can you say about im(A)? o (2 pits.) What can you say about a linear system whose coeflicient matrix is A? 4. Let T : 1R3 —) R3 be orthogonal projection to the subspace of R3 with basis consisting of the 2 single vector [ -—1 J . [Note: this is act a unit vector!] 0 (a) (4 pts.) Calculate Té’l, Té’g, Té'3. (b) (2 fits.) Find the matrix for T. (c) (4 pts.) What are the dimensions of ker(T) and im(T)? Find bases for these subspaces. You must justify your answers to receive full credit. ...
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