1.4 Linear Transformation I

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1 .4. LINEAR TRANSFORMATION I.4 Linear Transformation I MATH 9 FALL 99 PRELIM # 5 9FA9PQ5.tex.4. a) Consider the vector transformation y f(x) from V to V such that if y (y ; y ); x (x ; x ); y (x + x ) p y (x + x ) p : Verify that y A(x) is linear and nd a matrix A such that f(x) Ax for all x in V : b) Consider the linear transformation z g(y) from V to V with matrix " p p # B p p Find a matrix for the composite transformation z g(f(x)) ("function of a function"). MATH 9 FALL 99 FINAL # 5 9FA9FQ5.tex.4. Let T be the linear transformation T d dx acting on the space spanned by B f; sin (x); cos (x)g: a) Find the matrix T B which represents T in the basis B. b) If B is the basis B f; sin (x) + cos (x); sin (x) cos (x)g; nd the matrix T B, which represents T in the basis B. MATH 9 FALL 99 FINAL # 6 9FA9FQ6.tex.4. Consider the linear transformation, T, of the plane to itself, which is represented, in the standard basis, bythe non-singular matrix a b : c d Thus, x y a c b d x y The equation of a certain curve, (straight line), in the (x; y)-coordinate system is given by y mx + h. a) Find the equation of this same curve in the (x ; y )-coordinate system. b) What is the shape of this curve in the (x ; y )-coordinate system? :

2 MATH 9 SPRING 99 FINAL # 9SP9FQ.tex.4.4 Consider physical vectors v ai + bj + ck where a; b and c are scalars and i; j and k are mutually perpendicular unit vectors. A linear transformation is dened as T (v) (i j) x v Find the matrix representation of T in the (i; j; k) basis. MATH 9 SUMMER 99 FINAL # 4 9SU9FQ4.tex.4.5 Consider the vector space V with the standard basis B S (i; j; k): Now consider a second basis B which is obtained by rotating the basis B by degrees (anticlockwise) about the z axis. Also, consider the linear transformation T : V! V which reects any vector vv about the x-z plane. a) Find(B : B ). b) Find p matrix representations of T B and T B of T in the bases B and B. Hint: i + j is a vector in B. Also, check if (B : B ) (B : B ) t : MATH 9 SUMMER 99 FINAL # 5 9SU9FQ5.tex.4.6 Every vector ~v in two dimensional physical space can be written as ~v x^i + y^j where ^i and ^j are unit vectors on the positive x and y axes respectively. In each of the following cases, nd the matrix representing the linear transformation indicated and state whether or not it is invertible. a) T is the transformation which reects each vector about the y axis. b) T is the transformation which rotates each vector about the origin by an angle c) of 6 o in a counterclockwise direction. T is the transformation which transforms each vector into its vector projection on the x axis. MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.4.7 Consider linear transformation in < and the standard basis ; a) Find the matrix U of the linear transformation that stretches the x component of each vector by a factor of and keeps the y component unchanged. b) Find the matrix R of the linear transformation that rotates each vector by 45 degrees in the counterclockwise direction. c) Do the above transformations commute, i.e. is RU equal to UR? d) If yes, stop. If no, nd the matrix V such that RU V R

3 .4. LINEAR TRANSFORMATION I MATH 9 SPRING 99 FINAL # 7 9SP9FQ7.tex.4.8 Consider linear transformations L : V! V; where V is the vector space of all real x matrices, and consider the specic transformation dened by L(A) A A T () B where AV is a x real matrix and A T is the transpose of A. a) Show that L, as dened in () above, is a linear operator (transformation). b) Now consider N(L), the null space of L the set of all x matrices B such that L(B). Check the following matrices to see if they are in the null space of A ; B c) Find the null space of 4 A ; B MATH 9 FALL 994 FINAL # 9FA94FQ.tex.4.9 Which of the following transformations is linear? a a) L a b) L a c) L a d) L a e) L a a a a + + MATH 9 SPRING 995 PRELIM # 9SP95PQ.tex.4. a) Are the vectors ; 5 ; linearly independent? Do they span <? b) Are the vectors 4 5 ; ; 4 linearly independent? Do they span <? c) Let C(<) be the vector space of continuous functions on <. Are the three elements sin (x); sin (x + ); sin (x + ) linearly independent? Do they span C(<)? 5 A

4 4 MATH 9 FALL 995 PRELIM # 9FA95PQ.tex.4. Consider the linear transformation T : <! < that reects vectors through the plane y z. (You can think of the plane as a two-sided mirror.) a) Find the standard matrix A of T. You may use the fact that A [T (e); T (e); T (e)] if you wish. b) Is the transformation onto <? Give reasons for your answer. c) Is the transformation one-to-one? Give reasons for your answer. d) Determine A. Explain your result in geometric or physical terms. MATH 9 FALL 995 FINAL # 4 9FA95FQ4.tex.4. a) Show that translation in R! R, i.e. x x + h T () x x + k is not a linear transformation. b) Now consider homogeneous coordinates 4 x + h x + k 5. 4 x x 5 for x x with 4 x x 5 Find the x matrix for T. c) The shear transformation S : <! < along the x axis has the following matrix tan This transformation rotates the vector by an angle. Find this angle. d) Do the transformations T and S, commute in general? What happens in the special case k? Give reasons for your answer. MATH 9 SPRING 996 PRELIM # 9SP96PQa.tex.4. The following matrices apply to the next questions: A ; 4 A ; C 4 a) Is B, viewed as a linear transformation, one-to-one? If yes, explain why; if no, explain why and nd the solution set of Bx. MATH 9 SPRING 996 PRELIM # 9SP96PQa.tex.4.4 a) Is A, viewed as a linear transformation, onto? Explain why or why not. C A : A

5 .4. LINEAR TRANSFORMATION I 5 MATH 9 SPRING 996 PRELIM # 7 9SP96PQ7.tex.4.5 Which of the following functions are linear transformations? i) f : <! < ; f(x) x ii) g : <! < ; g(x) x + iii) S : <! < x ; S(x) x + iv) T : <! < 4 ; T (x) Cx, where C is the matrix above (in question ). v) R : < n! < m ; R(x) for all x MATH 9 SPRING 996 FINAL # 6 9SP96FQ6.tex.4.6 Let T : < 5! < be a linear transformation given by T (x) Ax, where A is a matrix and x is a vector in < 5. Then A has dimensions (rows, columns): a) x 5 b) 5 x c) 5 x 5 d) x e) none of the above MATH 9 SPRING 996 FINAL # 7 9SP96FQ7.tex.4.7 Let A : The best description of x! Ax is: a) a rotation about the origin. b) a reection through the x-axis. c) a reection through the y-axis. d) a reection through the origin. e) a reection through the line y x. : Then the ma- MATH 9 SPRING 996 FINAL # 9SP96FQ.tex Suppose that matrix A sends to and to 7 trixfor A is: 4 a) 8 8 b) 4 8 c) 4 8 d) 4 e) none of the above MATH 9 SPRING 996 FINAL # 9SP96FQ.tex.4.9 If A is 5 x 4, then x! Ax cannot map < 4 onto < 5. True or false. MATH 94 SPRING 997 PRELIM # 9 94SP97PQ9.tex.4. Suppose T : < n! < m is linear, and suppose that T (~x ) ~ b and T (~x ) ~ b. Find a vector ~x < n such that T (~x) : ~ b : ~ b :

6 6 MATH 94 SPRING 997 PRELIM # 4 94SP97PQ4.tex.4. Which of the following functions are linear transformations? You do not need to explain your answers. i) T : <! < ; given by x + y T x y y + ii) T : <! < ; T is reection in line y x +. iii) T : <! < ; T is reection in line x. iv) T 4 : <! <, given by T 4 (x) 4 cos ( 7 )x x x cos ( 7 ) MATH 94 SPRING 997 PRELIM # 5 94SP97PQ5.tex.4. Find the matrix A for the linear transformation T : <! < which is the composition of rst applying a rotation by angle clockwise, followed by then applying reection in the line y x. MATH 94 SPRING 997 PRELIM # 6 94SP97PQ6.tex.4. Find the matrix A for the linear transformation given by 4 x y z and nd the inverse of A. 5A 4 z x y MATH 94 SPRING 997 FINAL # 94SP97FQ.tex.4.4 (All parts are independent problems.) a) If the det A. Find the det A ; det A T. b) From P A LU nd a formula for A in terms of P; L and U. Assume P; L; U; A are invertible n x n matrices. c) Find the rank of matrix A: A [ ]: 5 ; 5 d) Find x matrix E such that for every x matrix A, the second row of EA is equal to the sum of the rst two rows of A, e.g., if A then EA e) Write down x matrix P which projects every vector on to x axis. Verify that P P.

7 .4. LINEAR TRANSFORMATION I 7 MATH 94 SPRING 997 FINAL # 7. 94SP97FQ7p.tex.4.5 Suppose A is a 6 row by 7 column matrix for which NulA Spanf~x o g for some ~x o 6 ~ in < 7 : Which of the following are always TRUE of A? (NO Justication is necessary.) Express your answers as e.g: TRUE: a,b,c,d; FALSE: e a) The columns of A are linearly dependent. b) The linear transformation ~x! A~x is onto. c) A~x ~ has only the trivial solution. d) The columns of A form a basis for < 6. e) The columns of A span all of < 6. MATH 94 FALL 997 PRELIM # 6 94FA97PQ6.tex.4.6 Let V be the vector space of x matrices. a) Find a basis for V. b) Determine whether the following subsets of V are subspaces. If so, nd a basis. If not, explain why not.. fa in V j det A g. fa in V ja A g: c) Determine whether the following are linear transformations. Give a short justication for your answers.. T : V! V; where T (A) A T ;. T : V! < ; where T (A) deta; MATH 94 SPRING 998 PRELIM # 4 94SP98PQ4.tex.4.7 a) Determine the x matrix that corresponds to a clockwise rotation through an angle. b) Using homogeneous coordinates nd the x matrix that describes the D composite transformations of reecting in the y axis and then translating (,). MATH 94 FALL 998 PRELIM # 94FA98PQ.tex.4.8 a) A transformation, T : < n! < n is dened as T (u) u + s; s constant vector. Is this a linear transformation? Why or why not? b) Sketch the image of the unit box, drawn below, after being mapped by the transformation x! Ax; A Clearly label on your sketch the images of the points labeled a, b, c, and d. Give a geometric interpretation of this transformation in words. :

8 8 MATH 94 SPRING 999 PRELIM # 94SP99PQ.tex.4.9 A linear transformation T : <! < maps the square shown (ABCD) to the parallelogram shown (A'B'C'D'). (The answers to the questions below do not depend on each other. You will not get credit for an incorrect answer to one part based on an incorrect answer to another part.) a) Find the matrix A so that T (x) Ax: b) Is the map one to one (why or why not)? c) Is the map onto (why or why not)? d) Describe in words the geometry of the transformation T (x) AAx: (Use one or more words, like 'stretch', 'rotate', 'reect', 'expand', 'project', 'shear' or 'translate' and describe the amount and/or orientation of such distortion.)