Math111 Hw
Math 111 Homework 1 fall 2007 due 14/9 1. (1.2; 17) Determine the values of h such that the matrix is the augmented matrix of a system which admits a solution. 2 3 4 6 h 7
2. (1.2; 12) Find the general solutions of the system whose augmented matrix is 1 −7 0 6 5 0 0 1 −2 −3 −1 7 −4 2 7 1 −2 4 3. (1.3; 17) Let a1 = 4 , a2 = −3 , b = 1 . For what −2 7 h value(s) of h is b in the plane spanned by a1 and a2? 4. (1.4; 15) Let A = b1 2 −1 and b = . Show that the equation −6 3 b2 Ax = b does not have a solution for all possible b, and describe the set of all b for which Ax = b does have a solution.
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Math 111 Homework 2 spring 2007 due 21/9 1. (1.5; 26) Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. 2. (1.7; 7) Determine if the columns of the matrix form a linear independent set by the definition of linear independence. 1 4 −3 0 −2 −7 5 1 −4 −5 7 5 3. (1.7; 10) Let 1 −2 2 v1 = −5 , v2 = 10 , v3 = −9 −3 6 h (a) For what values of h is v3 in span{v1 , v2}? (b) For what values of h is {v1 , v2, v3} a linearly dependent set? 4. (1.8; 24) Suppose vectors v1 , ..., vp span Rn, and let T : Rn → Rn be a linear transformation. Suppose T (vi ) = 0 for all i = 1, ..., p. Show that T is the zero transformation. That is, show that if x is any vector in Rn , then T (x) = 0.
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Math 111 Homework 3 spring 2007 due 28/9 x1 7 −2 , and let , and v2 = , v1 = −3 x2 5 2 2 T : R → R be a linear transformation that maps x into x1v1 + x2v2 . Find a matrix A such that T (x) = Ax for each x.
1. (1.8; 20) Let x =
2. (1.9; 8) Find the standard matrix of T : R2 → R2 if T (x) is obtained first by reflecting x through the horizontal x1−axis and then reflects through the line x1 = x2. (Assume that T is linear.) 3. (1.9; 12) Show that the transformation T in the previous problem is merely a rotation about the origin. What is the angle of rotation? 4. (1.9;
2. (1.2; 12) Find the general solutions of the system whose augmented matrix is 1 −7 0 6 5 0 0 1 −2 −3 −1 7 −4 2 7 1 −2 4 3. (1.3; 17) Let a1 = 4 , a2 = −3 , b = 1 . For what −2 7 h value(s) of h is b in the plane spanned by a1 and a2? 4. (1.4; 15) Let A = b1 2 −1 and b = . Show that the equation −6 3 b2 Ax = b does not have a solution for all possible b, and describe the set of all b for which Ax = b does have a solution.
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Math 111 Homework 2 spring 2007 due 21/9 1. (1.5; 26) Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. 2. (1.7; 7) Determine if the columns of the matrix form a linear independent set by the definition of linear independence. 1 4 −3 0 −2 −7 5 1 −4 −5 7 5 3. (1.7; 10) Let 1 −2 2 v1 = −5 , v2 = 10 , v3 = −9 −3 6 h (a) For what values of h is v3 in span{v1 , v2}? (b) For what values of h is {v1 , v2, v3} a linearly dependent set? 4. (1.8; 24) Suppose vectors v1 , ..., vp span Rn, and let T : Rn → Rn be a linear transformation. Suppose T (vi ) = 0 for all i = 1, ..., p. Show that T is the zero transformation. That is, show that if x is any vector in Rn , then T (x) = 0.
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Math 111 Homework 3 spring 2007 due 28/9 x1 7 −2 , and let , and v2 = , v1 = −3 x2 5 2 2 T : R → R be a linear transformation that maps x into x1v1 + x2v2 . Find a matrix A such that T (x) = Ax for each x.
1. (1.8; 20) Let x =
2. (1.9; 8) Find the standard matrix of T : R2 → R2 if T (x) is obtained first by reflecting x through the horizontal x1−axis and then reflects through the line x1 = x2. (Assume that T is linear.) 3. (1.9; 12) Show that the transformation T in the previous problem is merely a rotation about the origin. What is the angle of rotation? 4. (1.9;