Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Singular-value decomposition wikipedia , lookup
Determinant wikipedia , lookup
History of algebra wikipedia , lookup
Matrix calculus wikipedia , lookup
Non-negative matrix factorization wikipedia , lookup
System of linear equations wikipedia , lookup
Cayley–Hamilton theorem wikipedia , lookup
Linear algebra wikipedia , lookup
Orthogonal matrix wikipedia , lookup
Linear Algebra Problem Set 2.3~2.4 2.3 Elimination Using Matrices (27) 27. Choose the numbers a, b, c, d in this augmented matrix so that there is: (a) No solution (b) Infinitely many solutions (c) Which of the numbers a, b, c, or d have no effect on the solvability? ⎡1 2 3 a ⎤ [ A b] = ⎢⎢0 4 5 b ⎥⎥ ⎢⎣0 0 d c ⎥⎦ Sol. (a) d = 0 and c ≠ 0 (b) c = 0 and d = 0 (c) a and b Recitation 2 ‐ 2008/09/30 1 Linear Algebra Problem Set 2.3~2.4 2.4 Lengths and Dot Products (2.4 C, 10, 26, 27, 33, 34) 2.4 C A directed graph starts with n nodes. There are n2 possible edges-each edge leaves one of the n nodes and enters one of the n nodes (possibly itself). The n by n adjacency matrix has aij = 1 when an edge leaves node i and enters node j; if no edge then aij = 0. Here are two directed graphs and their adjacency matrices: ⎡1 1 ⎤ A=⎢ ⎥ ⎣1 0⎦ ⎡1 1 1 ⎤ ⎢1 0 0⎥ ⎢ ⎥ ⎢⎣1 0 0⎥⎦ (a) List all of the 3-step paths between each pair of nodes and compare with A3. (b) What is the diameter of the second graph? Sol. ⎡3 2⎤ A3 = ⎢ ⎥ ⎣2 1 ⎦ (a) ⎡1 to 1 to 1 to 1, 1 to 2 to 1 to 1, 1 to 1 to 2 to 1 ⎢ 2 to 1 to 2 to 1, 2 to 1 to 1 to 1 ⎣ 1 to 1 to 1 to 2, 1 to 2 to 1 to 2⎤ ⎥ 2 to 1 to 1 to 2 ⎦ ⎡1 1 1 ⎤ ⎡1 1 1 ⎤ ⎡3 1 1⎤ (b) A = ⎢⎢1 0 0 ⎥⎥ ⎢⎢1 0 0 ⎥⎥ = ⎢⎢1 1 1⎥⎥ , the second graph has diameter 2. ⎢⎣1 0 0⎥⎦ ⎢⎣1 0 0 ⎥⎦ ⎢⎣1 1 1⎥⎦ 2 10. Row 1 of A is again added to row 2 to produce EA. Then F adds row 2 of EA to row 1. The result is F(EA): b ⎤ ⎡ 2a + c 2b + d ⎤ ⎡a b ⎤ ⎡1 1⎤ ⎡ a A=⎢ , F ( EA) = ⎢ ⎥ ⎥ ⎢ ⎥=⎢ ⎥ ⎣c d ⎦ ⎣0 1⎦ ⎣ a + c b + d ⎦ ⎣ a + c b + d ⎦ (a) Do those steps in the opposite order: first add row 2 to row 1 by FA, then add row 1 of FA to row 2. (b) What law is or is not obeyed by matrix multiplication? Sol. ⎡1 0⎤ ⎡ a + c b + d ⎤ ⎡ a + c b + d ⎤ = (a) E ( FA) = ⎢ ⎥⎢ d ⎥⎦ ⎢⎣ a + 2c b + 2d ⎥⎦ ⎣1 1 ⎦ ⎣ c (b) Commutative law Recitation 2 ‐ 2008/09/30 2 Linear Algebra Problem Set 2.3~2.4 26. Multiply AB using columns times rows: ⎡1 0 ⎤ ⎡1 ⎤ ⎡3 3 0 ⎤ ⎢ ⎥ ⎢ ⎥ AB = ⎢ 2 4⎥ ⎢ = 2 [ 3 3 0] + 1 2 1 ⎥⎦ ⎢ ⎥ ⎣ ⎢⎣ 2 1 ⎥⎦ ⎢⎣ 2 ⎥⎦ = Sol. ⎡1 AB = ⎢⎢ 2 ⎢⎣ 2 ⎡3 = ⎢⎢6 ⎢⎣6 0⎤ ⎡1 ⎤ ⎡0 ⎤ ⎡3 3 0 ⎤ ⎢ ⎥ ⎥ = ⎢ 2⎥ [3 3 0] + ⎢⎢ 4 ⎥⎥ [1 2 1] 4⎥ ⎢ ⎥ 1 2 1⎦ ⎢⎣ 2⎥⎦ ⎢⎣1 ⎥⎦ 1 ⎥⎦ ⎣ 3 0⎤ ⎡0 0 0 ⎤ ⎡ 3 3 0 ⎤ 6 0⎥⎥ + ⎢⎢ 4 8 4⎥⎥ = ⎢⎢10 14 4⎥⎥ 6 0⎥⎦ ⎢⎣1 2 1 ⎥⎦ ⎢⎣ 7 8 1 ⎥⎦ 27. The product of upper triangular matrices is always upper triangular: ⎡ x x x⎤ ⎡ x x x⎤ ⎡ AB = ⎢⎢ 0 x x ⎥⎥ ⎢⎢ 0 x x ⎥⎥ = ⎢⎢0 ⎢⎣ 0 0 x ⎥⎦ ⎢⎣ 0 0 x ⎥⎦ ⎢⎣0 0 ⎤ ⎥ ⎥ ⎥⎦ (a) Row times column is dot product (Row 2 of A)⋅ (Column 1 of B) = 0. Which other dot products give zeros? (b) Column times row is full matrix Draw x’s and 0’s in (column 2 of A) times (row 2 of B) and in (column 3 of A) times (row 3 of B). Sol. (a) (Row 3 of A)⋅ (Column 1 of B) and (Row 3 of A)⋅ (Column 2 of B) ⎡ x⎤ ⎡0 x x ⎤ ⎡ x⎤ ⎡0 0 x ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x [ 0 0 x ] = ⎢⎢0 0 x ⎥⎥ (b) ⎢ x ⎥ [ 0 x x ] = ⎢0 x x ⎥ , ⎢ ⎥ ⎢⎣ 0 ⎥⎦ ⎢⎣0 0 0 ⎥⎦ ⎢⎣ x ⎥⎦ ⎢⎣0 0 x ⎥⎦ Recitation 2 ‐ 2008/09/30 3 Linear Algebra Problem Set 2.3~2.4 33. Suppose you solve Ax=b for three special right sides b: ⎡1 ⎤ ⎡0 ⎤ ⎡0 ⎤ ⎢ ⎥ ⎢ ⎥ Ax1 = ⎢0 ⎥ , Ax2 = ⎢1 ⎥ , Ax3 = ⎢⎢0⎥⎥ ⎢⎣0 ⎥⎦ ⎢⎣0⎥⎦ ⎢⎣1 ⎥⎦ If the three solutions x1, x2, x3 are the columns of a matrix X, which is A times X? Sol. AX = [ Ax1 Ax2 Ax3 ] 34. If the three solutions in Question 33 are x1=(1, 1, 1) and x2=(0, 1, 1) and x3=(0, 0, 1), solve Ax=b when b=(3, 5, 8). Challenge problem: What is A? Sol. ⎡1⎤ ⎡1 ⎤ ⎡0⎤ ⎡0⎤ ⎡0⎤ ⎡0 ⎤ A ⎢⎢1⎥⎥ = ⎢⎢0⎥⎥ , A ⎢⎢1 ⎥⎥ = ⎢⎢1 ⎥⎥ , A ⎢⎢0⎥⎥ = ⎢⎢0 ⎥⎥ ⎢⎣1⎥⎦ ⎢⎣0⎥⎦ ⎢⎣1 ⎥⎦ ⎢⎣0⎥⎦ ⎢⎣1 ⎥⎦ ⎢⎣1 ⎥⎦ ⎡1 0 0 ⎤ ⎡1 0 0⎤ ⎡ 1 0 0⎤ ⎢ ⎥ ⎢ ⎥ A ⎢1 1 0 ⎥ = ⎢0 1 0⎥ , A = ⎢⎢ −1 1 0⎥⎥ ⎢⎣1 1 1 ⎥⎦ ⎢⎣0 0 1 ⎥⎦ ⎢⎣ 0 −1 1 ⎥⎦ Recitation 2 ‐ 2008/09/30 4