University of California San Diego NAME TA: Linear Algebra Wednesday, October st, 9 :am - :5am No aids are allowed Be sure to write all row operations used Remember that you can often check your answers Answers without full justification are worth no credit This test consists of 4 questions Each question is worth a total of 5 points Page of 6
Let A = 3 5 5, b = (a) Find the full solution set of the nonhomogeneous system Ax = b, written in parametric vector form [3] 4 We construct the augmented matrix associated with the system and perform the row reduction algorithm, 3 4 5 5 3 4 R + R 5 R 3 + R R R ( ) 3 4 5 5 3 4 R 3 + R ( ) R R Using the reduced Echelon form obtained above, we can write the following system of equations, x + x 3 = x x 3 = Hence, x x 3 x = x = + x 3 = + x 3 x 3 x 3 is the solution of the above system written in parametric vector form (b) What is the parametric vector form for the solution of the corresponding homogeneous system Ax = [] Recall that the solution set of Ax = b is the set of all vectors of the form w = p+v h, where p is a particular solution of the nonhomogeneous system Ax = b and v h is any solution of the corresponding homogeneous system Ax = It is easy to see that p = [ ] T is a solution of the nonhomogeneous system Ax = b Consequently, the solution set for the homogeneous system Ax = is x = x x x 3 = x 3 Page of 6
(c) Write the vector b as a linear combination of the columns of the matrix A [] Let A = [a a a 3 ], then by definition of matrix-vector multiplication, b = Ax = x a + x a + x 3 a 3 In order to write b as a linear combination of the columns of the matrix A, it is sufficient to choose some x from the solution set found in a) For example, let x 3 =, then x = + ( ) = Substituting this x into the system leads to 4 b = = Ax = 5 + + ( ) 3 5 Page 3 of 6
Let V = {[ 4 8 ] [ 6, ]} (a) Is the set V linearly independent? Justify your answer [] The set is not linearly independent, because one of the vectors is a multiple of another In particular, [ ] 4 = [ ] 6 8 3 In other words, there exist non-zero weights c =, c = /3, such that [ ] 4 + [ ] [ ] 6 =, 8 3 which by definition means that the vectors are linearly dependent (b) For v = (4, 8), v = ( 6, ), give a definition of Span{v, v } [] For v, v R, the set of all linear combinations of v, v is defined to be a span of v, v That is, Span{v, v }, is the collection of all vectors, with c, c scalars c v + c v, (c) Give a geometric description of Span{v, v } (ie, draw a picture and give explanation) [] Since the set of vectors {v, v } is linearly dependent, any linear combination c v + c v degenerates to a linear combination of a single vector In more detail, c v + c v = (c 3 c )v Since c, c are arbitrary scalars, we can write the span as just γv, where γ is any scalar Geometrically, this corresponds to a line through the origin 5 6 4 4 6 5 Page 4 of 6
3 Consider T : R R 3, T (x, x ) = (x x, x + 3x, x + x ) (a) Show that T is a one-to-one linear transformation Does T map R onto R 3? Justify your answer [3] We can determine the standard matrix of T by inspection, x x [ T (x) = x + 3x = 3 x x x + x One could also use the definition, by which the matrix of T is constructed as [T (e ) T (e )], where [e e ] is the identity matrix Since T (x) = Ax, where A = 3, the transformation is linear The columns of the matrix A are not multiples of each other, so they are linearly independent By the theorem from class (Theorem, Section 9), this is equivalent to the transformation T being one-to-one By the same theorem, the transformation is onto, if the columns of A span R 3 The latter is not possible because two vectors in R cannot span the whole space Therefore, the transformation T is not onto ] (b) Consider S : R 3 R, S(x, x, x 3 ) = (x + x 3, x x 3 ) Compute the composition transformation x S(T (x)) (from R to R ) Find the standard matrix for S(T (x)) [] For convenience, let us write S(y, y, y 3 ) = (y + y 3, y y 3 ) The the composition is, S(T (x, x )) = S(x x, x + 3x, x + x ) = ((x x ) + (x + x ), (x x ) (x + x )) = (x, x ) It remains to note that [ x S(T (x, x )) = x ] = [ ] [ x x ] Page 5 of 6
4 Let A = 4 7 3 6 3 (a) Find A [3] In order to obtain A, we row-reduce the augmented matrix [A 4 7 3 6 3 4 3 6 Hence, R R 4 R 3 + R R + R 3 R R 3 A = 4 9 6 3 6 I] R 3 R R + R (b) Explain why Ax = has only the trivial solution [] There are many ways to answer this question One possible argument is due to Theorem 5 in Section By this theorem, if A is invertible, then for any b R 3, the equation Ax = b has the unique solution x = A b In particular, for b =, the unique solution of the homogeneous system, Ax =, is x = A = (the latter follows easily from the definition of matrix-vector multiplication), which is the trivial solution (c) What is (A T )? [] Here we use the fact that (A T ) = (A ) T So 3 6 (A T ) = Page 6 of 6