Honors Advanced Math More on Determinants, Transformations and Systems 14 May 2013

Transcription

1 Honors Advanced Math Name: More on Determinants, Transformations and Sstems 14 Ma 013 Directions: The following problems are designed to help develop connections between determinants, sstems of equations and matrix transformations 1 Suppose T a ab ac abc a Do a calculation to show that det T 0 Notice that ou could have known this just b looking at the rows and columns of T b Consider the rows of T Suppose u r a,ab and v r ac,abc Notice that these vectors are parallel Find a number k such that k u v c Now consider the columns of T Suppose u c a,ac and v c ab,abc Notice that these vectors are parallel Find a number k such that k u v d Use T to find the image of the points ( 1,1 ) and ( 5, 1) e Explain how ou can tell that the transformation T maps all points in the x-plane to a single line Then find the equation of that line Suppose T a Do a calculation to show that det T 0 Notice that ou could have known this just b looking at the rows and columns of T b Consider the rows of T Suppose u r, 6 and v r 5, 15 Notice that these vectors are parallel Find a number k such that k u v c Now consider the columns of T Suppose u c, 5 and v c 6, 15 Notice that these vectors are parallel Find a number k such that k u v d Use T to find the image of the points ( 1,1 ) and ( 5, 1) e Explain how ou can tell that the transformation T maps all points in the x-plane to a single line Then find the equation of that line f You should have found that both of the pre-image points in the last problem mapped to the point ( 4,10) Find the set of all points that map to (4, 10) g Find the set of all points that map to the origin h Find the set of all points that map to (-, -5) i You ma have noticed that the points (4, 10), (0, 0), and (-, -5) are collinear Find the equation of this line j Look at the rows and columns of T You ma have noticed that the row 1 and row are in a :5 ratio and that column 1 and column are in a 1:3 ratio How do these observations relate to our answers to parts f and i? k Graph the equations that ou found in f, g, h, and i on the grid at right What do ou notice about the mappings? Does (4, 10) map to (4, 10)?

2 More on Determinants, Transformations and Sstems page Suppose T a Do a calculation to show that det T 0 b Consider the rows of T Suppose u r 1, 1,, v r 3,0,1, and w r 7,, 1 These vectors are not parallel (how can ou tell?), but the do form a linear combination In other words, there are numbers, k 1 and k such that k 1 u + k v w Find k 1 and k c Now consider the columns of T Suppose u c 1,3,7, v c 1,0,, and w c,1, 1 These vectors are not parallel (how can ou tell?), but the do form a linear combination Find c 1 and c such that c 1 u + c v w d The transformation T maps all points in 3-space into a single plane 1 Find the equation of this plane e Use T to find the image of the points ( 1, 1, 1) and (, 4, ) f You should have found that both of the pre-image points in the last problem mapped to the point (,4,8) Find the set of all points that map to (,4,8) g Find the set of all points that map to the origin h How does our answer to the last problem relate to our answers to parts b and c? 4 The intersection of two non-parallel planes is a line Find the equation of the line Give the equation in parametric form and/or in vector form Hint: Use RREF, then let t z 1st pair of planes M : x z 3 N : 4 x + 13z 11 nd pair of planes M : 5x 9 7z 17 N : 4 x 5 10z 7 5 In the next three problems, ou are given the equations of three planes Determine the intersection of all three planes (this could be a point, a line, a plane or no intersection at all) If the intersection of all three is a line, find an equation for the line Hint: Before doing the problems, determine whether an of the planes are parallel or the same Then use RREF 1st set of three planes nd set of three planes 3rd set of three planes m 1 : x + z 3 m 1 : x 3 + z 1 m 1 : 3x 5x + 6z 7 m : 5x + 4z 7 m 3 : x + 4 z 5 m : 3x 8 + 3z 5 m 3 : x 5 + z 4 m : x - + 5z -6 m 3 : x z -9 1 You might be wondering how we can tell that T doesn t map everthing to a single line Here s a wa to think about it In part c, we noticed that the third column was a linear combination of the first two columns This means that the vector w is in the plane formed b u and v We know that u and v form a plane because u and v are not parallel We have three vectors, but we are short one dimension, so 3-space gets mapped to a plane If u, v, and w were all parallel, then everthing would map to a line (a loss of two dimensions)

3 More on Determinants, Transformations and Sstems page 3 6 Consider the nd set of planes from the last problem Observe that this problem is equivalent 1 3 x 1 to solving the matrix equation z 4 a Show that the determinant of the transformation matrix in this problem is 0 b The transformation matrix in this problem maps everthing to a plane Find the equation of the plane (Hint: What does this matrix do to i, j, and k?) c Is the point ( 1,5,4 ) in the plane from part b? Wh is this significant? d Find the set of all points that are mapped to ( 1,5,4 ) Wh is this the same as the answer ou got for the second part of problem 5? Review Problems Directions: Here are some extra problems on matrices The var in terms of length and difficult Some are problem tpes that we've alread done and some are a little new All of them are within our reach In studing for our test, ou should also make sure ou understand the problems from the previous homework assignments and that ou know how to do all of the major proofs from the unit x 9 1 a Find the partial fraction decomposition of In other words, find A and B such x 3x 8 x 9 A B that + Note: If ou re stuck, see section 74 in Demana x 3x 8 x + 4 x 7 b Find the partial fraction decomposition of 11x + 35 x 5x 9x 3x 40 c Find the partial fraction decomposition of 3 x 3x 10x Each of the following matrices represents a sstem of equations Problems a c involve 3 variables and problem d has 4 Determine the solution set of each sstem If the solution set is a line, find the equation of the line If the solution set is a plane, find the equation of the plane a b c d

4 More on Determinants, Transformations and Sstems page 4 3 Consider the following sstem of equations k x k Suppose the following sstem of equations has a unique solution for x and What can ou conclude about the value of k? Consider the linear transformation 4 0 x ʹ x ʹ The effect of this transformation can be reached using two of the basic transformations (reflection, rotation, dilation, translation) Identif which two Be as specific as possible Give the matrix for each transformation Does the order of these transformations matter? 5 Suppose T is a transformation matrix that has the following effect It first rotates the preimage point 10 o clockwise, then reflects over the -axis, and then compresses verticall b a factor of Find T Consider the linear transformation 6 x ʹ x ʹ a How can ou tell that this transformation will map all of the points in the plane to a single line? b What is the equation of this line? 4, 8 c Find a point whose image is ( ) d Find another point whose image is ( 4, 8) e How could ou describe the set of all points whose image is( 4, 8)? f How can ou tell there is no point whose image is ( 9, 6)? 7 Consider the transformation x' x + + z, ' x + z, and z' 5 x z a Find the image of the point (, 4, 5) b Find a point whose image is the point ( 4, 7, 8) 8 Give a geometric description of the following transformations Be as specific as possible (For example, if the transformation is a reflection, tell what plane the points are being reflected over) If ou aren't sure what a transformation does, tr a few points and see if ou can identif a pattern a x' x b x' x + 3 c x' z ' ' 5 ' z' z z' z 1 z' x

5 More on Determinants, Transformations and Sstems page 5 9 Turn each verbal description into transformation equations (ax + b + cz + d xʹ, etc) a A transformation that shifts all of the z values up b 10, shifts the values down b 4 and shifts the x values left b 7 b A transformation that reflects all points over the z-plane c A transformation that rotates all points in the x-plane b 90 o counterclockwise d A transformation that rotates all points in the z-plane b 90 o clockwise 10 Rewrite the following to transformations using matrices Linear Transformation x' ax + b + cz ' dx + e + fz z' gx + h + iz Non-Linear Transformation x' x + h ' + k z' z + l 11 Multipl the following a b c 1 d e f 0 g h i 0 a b c 0 d e f 1 g h i 0 a b c 0 d e f 0 g h i 1 1 Find matrices for each of the following transformations a Reflect over the x-plane b Stretch the x values b 3, stretch the values b 5, and stretch the z values b c Rotate the z-plane b 90 o counterclockwise d Rotate the xz-plane b 30 o counterclockwise e Rotate the z-plane b θ o counterclockwise 1 3 x xʹ 13 Consider the linear transformation ʹ z zʹ a How can ou tell that the set of image points for this transformation will not be all of 3-space? b What is the set of all image points? (A plane? A line? Something else?) Find an equation for the set of all image points 14 Solve the matrix equation AX + B CX + D for X Your answer will be in terms of A, B, C, and D 15 Find the matrix of a linear transformation that reflects over the line x