Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns


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1 Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in R is just the set Span 2, 5 6, so the question is asking whether or not u lies in that set. As in previous exercises, this leads to a linear system whose augmented matrix is reduced to echelon form as: The system is consistent (since the last column is not a pivot column), so u does lie in the given plane.. Suppose A is a matrix and b is a vector in R with the property that A x = b has a unique solution. What can you say about the reduced echelon form of A? Justify your answer. Since A x = b has a unique solution, the associated linear system has no free variables, and therefore all columns of A are pivot columns. So the reduced echelon form of A must be. 6. Suppose A is a matrix and b is a vector in R with the property that A x = b has a unique solution. Explain why the columns of A must span R. As in the previous exercise, since A x = b has a unique solution, all columns of A must be pivot columns. Therefore there are four pivot columns, hence four pivot elements. These must all lie in different rows, so since there are four rows, all rows must contain a pivot element. By Theorem on page 9, it follows that the columns of A span R.
2 2 Section.5 (Page 9) 2. Let p = and q =. Find a parametric equation of the line M through p and q. Hint: M is parallel to the vector q p. See the figure below omitted].] q p =. The line containing this vector is Span{ q p}, and is given in parametric form as x = t ( t in R ). Therefore (as on page 7) the line through p and q is obtained by translating that line by p ; it is given in parametric form as x = + t ( t in R ). (You could also use q in place of p.) 7. Construct a 2 2 matrix A such that the solution set of the equation A x = is the line in R 2 through (, ) and the origin. Then, find a vector b in R 2 such that the solution set of A x = b is not a line in R 2 parallel to the solution set of A x =. Why does this not contradict Theorem 6? We can find a homogeneous linear equation in (x, x 2 ) that has solution x =, x 2 = ; it is x x 2 = (or any nonzero scalar multiple of this equation). We need a linear system with two such equations, so we can just use this equation twice. The coefficient matrix of this linear system is our matrix A : A =. For any vector x in R 2, the two entries of the product A x must be the same. So, let b =. Then the matrix equation A x = b is inconsistent, because when you row reduce the matrix A b ] you find that the last column is a pivot column. The solution set of this matrix equation is empty, so it is not a line in R 2 parallel to the solution set of A x =. This does not contradict Theorem 6, because Theorem 6 applies only to consistent equations, and this system is not consistent.
3 8. Let A be an m n matrix and let w be a vector in R n that satisfies the equation A x =. Show that for any scalar c, the vector c w also satisfies A x =. That is, show that A(c w) =.] By Theorem 5(b) (page ) and the fact that w satisfies A x =, Therefore c w satisfies the equation. A(c w) = c(a w) = c =. Section.7 (Page 62) 8. Determine if the columns of the matrix form a linearly independent set. Justify your answer. They are linearly dependent, by Theorem 8 on page 6 (the matrix has more columns than rows).. Find the value(s) of h for which the vectors 5, 2 9, 6 h 9 are linearly dependent. Justify your answer. Let us row reduce the matrix whose columns are these vectors: h h 5 The vectors are linearly dependent for all values of h, because the third column of the above matrix is never a pivot column. (This is similar to Example 2 on page 59, except that we left out the last column of the augmented matrix, since it is always zero and therefore does not affect the process of row reduction (it is never a pivot column).)
4 6. The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification. (One specific example cannot explain why a statement is always true.) If v, v 2, v are in R and v is not a linear combination of v and v 2, then { v, v 2, v } is linearly independent. Take v =, v 2 =, and v =. This (indexed) set is linearly dependent (because the first vector is zero), but one cannot write v as a linear combination of v and v 2, because the first coordinates of v and v 2 are zero, but the first coordinate of v is not zero.. Suppose an m n matrix A has n pivot columns. Explain why for each b in R m the equation A x = b has at most one solution. Hint: Explain why A x = b cannot have infinitely many solutions. The matrix A has n pivot columns, which is equal to its number of columns. Therefore every matrix of A is a pivot column. Therefore, in an augmented matrix A b ], all columns except for possibly the last one will be pivot columns since the pivots of the A part of this matrix are the same as the pivots of A. So the equation A x = b cannot have infinitely many solutions (regardless of what b is). Section.8 (Page 7). Find all x in R that are mapped into the zero vector by the transformation x A x for the matrix A =. 2 8 The question amounts to solving the matrix equation A x =, so we row reduce its augmented matrix to reduced echelon form:
5 In parametric vector form, the solutions are therefore 2 2 x = x + x. 2. Let b =, and let A be the matrix in Exercise. Is b in the range of the linear transformation x A x? Why or why not? This leads to a linear system whose augmented matrix is (partially) row reduced as follows: At this point we can stop, because it is clear that the last column is a pivot column, so the linear system is inconsistent. Therefore b is not in the range of x A x. (Another way to see this is to notice that if b is to equal the expression in the answer to Exercise, then x must be and x must be. But using those values 8 gives x =, which is not b.) x 7 2. Let x =, v x =, and v =, and let T : R 2 2 R 2 be a linear transformation that maps x into x v + x 2 v 2. Find a matrix A such that T ( x) is A x for each x. A = T ( e ) T ( e 2 ) ] = v v 2 ] = 7 ] Show that the transformation T defined by T (x, x 2 ) = (x 2 x 2, x x 2 ) is not linear. In this exercise, column vectors are written as rows; for example, x = (x, x 2 ) and T (x) is written as T (x, x 2 ).]. 5
6 6 but T (, ) + T (, ) = ( 2, ) + ( 2, ) = (, ), T ((, ) + (, )) = T (, ) = (, ). These are not equal, so T does not preserve vector addition. Therefore it is not a linear transformation. Section.9 (Page 8). Find the standard matrix of T, where T : R 2 R 2 first reflects points through the horizontal x axis and then reflects points through the line x 2 = x. T : x x 2 ] Therefore the standard matrix of T is x T ( e ) T ( e 2 ) ] = ] x2. x 2 x You can also see this using the standard matrices given in the first and third rows of the table on page 75: A = =. 2. Show that the transformation in Exercise is merely a rotation about the origin. What is the angle of the rotation? The standard matrix obtained in Exercise coincides with the matrix of Example on page 7 when φ = π/2. So the transformation is rotation counterclockwise about the origin by π/2 radians.. Let S : R p R n and T : R n R m be linear transformations. Show that the mapping x T (S( x)) is a linear transformation (from R p to R m ). Hint: Compute T (S(c u + d v)) for u, v in R p and scalars c and d. Justify each step of the computation, and explain why this computation gives the desired conclusion.] T (S(c u + d v)) = T (S(c u) + S(d v)) since S preserves addition = T (cs( u) + ds( v)) since S preserves scalar multiplication = T (cs( u)) + T (ds( v)) since T preserves addition = ct (S( u)) + dt (S( v)) since T preserves scalar multiplication. Taking c = d = gives T (S( u + v)) = T (S( u)) + T (S( v)), and taking d =, v = gives T (S(c u)) = ct (S( u)). These are the two properties required for x T (S( x)) to be a linear transformation..
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