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Math 3013
Homework Set 4
Problems from ยง1.6 (pgs. 99-101 of text): 1,3,5,7,9,11,17,19,21,24,26,35,37,
- (Problems 1.6.1, 1.6.3, 1.6.4, 1.6.7, 1.6.9 in text). Determine whether the indicated subset is a subspace of the given Rn.
(a) W = {[r, โr] | r โ R} in R^2
(b) W = {[n, m] | n and n are integers} in R^2
(c) W = {[x, y, z] | x, y, z โ R and z = 3x + 2} in R^3
(d) W = {[x, y, z] | x, y, z โ R and z = 1, y = 2x} in R^3
(e) W = {[2x 1 , 3 x 2 , 4 x 3 , 5 x 4 ] | xi โ R} in R^4
- (Problem 1.6.11 in text). Prove that the line y = mx is a subspace of R^2. (Hint: write the line as W = {[x, mx] | x โ R}.)
- (Problems 1.6.17, 1.6.19 and 1.6.21 in text). Find a basis for the solution set of the following homogeneous linear systems.
(a) 3 x 1 + x 2 + x 3 = 0 6 x 1 + 2x 2 + 2x 3 = 0 โ 9 x 1 โ 3 x 2 โ 3 x 3 = 0
(b) 2 x 1 + x 2 + x 3 + x 4 = 0 x 1 โ 6 x 2 + x 3 = 0 3 x 1 โ 5 x 2 + 2x 3 + x 4 = 0 5 x 1 โ 4 x 2 + 3x 3 + 2x 4 = 0
(c)
x 1 โ x 2 + 6x 3 + x 4 โ x 5 = 0 3 x 1 + 2x 2 โ 3 x 3 + 2x 4 + 5x 5 = 0 4 x 1 + 2x 2 โ x 3 + 3x 4 โ x 5 = 0 3 x 1 โ 2 x 2 + 14x 3 + x 4 โ 8 x 5 = 0 2 x 1 โ x 2 + 8x 3 + 2x 4 โ 7 x 5 = 0
- (Problems 1.6.35 and 1.6.37 in text). Solve the following linear systems and express the solution set in a form that illustrates Theorem 1.18.
(a)
2 x 1 โ x 2 + 3x 3 = โ 3 4 x 1 + 4x 2 โ x 4 = 1 1
2
(b) 2 x 1 + x 2 + 3x 3 = 5 x 1 โ x 2 + 2x 3 + x 4 = 0 4 x 1 โ x 2 + 7x 3 + 2x 4 = 5 โx 1 โ 2 x 2 โ x 3 + x 4 = โ 5
- (Problem 1.6.38 in text). Mark each of the following statements True or False.
a. A linear system with fewer equations than unknowns has an infinite number of solutions.
b. A consistent linear system with fewer equations than unknowns has an infinite number of solutions.
c. If a square linear system Ax = b has a solution for every choice of column vector b, then the solution is unique for each choice of b.
d. If a square system Ax = 0 has only the trivial solution x = 0 , then Ax = b has a unique solution for every column vector b with the appropriate number of components.
e. If a linear system Ax = 0 has only the trivial solution x = 0 , then Ax = b has a unique solution for every column vector b with the appropriate number of components.
f. The sum of two solution vectors of any linear system is also a solution vector of the system.
g. The sum of two solution vectors of any homogeneous linear system is also a solution vector of the system.
h. A scalar multiple of a solution vector of any homogeneous linear system is also a solution vector of the system.
i. Every line in R^2 is a subspace of R^2 generated by a single vector.
j. Every line in R^2 through the origin in R^2 is a subspace of R^2 generated by a single vector.
