Solutions Week 2. D-ARCH Mathematics Fall Linear dependence & matrix multiplication. 1. A bit of vector algebra.

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1 D-ARCH Mathematics Fall 4 Solutions Week Linear dependence & matrix multiplication A bit of vector algebra (a) Find all vectors perpendicular to Drawing a sketch might help! Solution : All vectors in the plane described b the equation x+ z = (b) Find all solutions x, x, x of the equation b = x v + x v + x v, where 8 4 b =, v = 4 7, v = 5 8, v = Solution : The linear sstem has unique solution x =, x =, x = 4 (c) Are v, v, v linearl dependent? Solution : No The equation x v + x v + x v = has unique solution Consider the three vectors v, v, v in the x--plane : v v Are v, v linearl dependent? What about v, v, v? Argue geometricall v Solution : The vectors v, v are linearl dependent if and onl if one is a scalar multiple of the other But if v were a scalar multiple of v, it would have to lie along the line going through v In the picture, this is clearl not the case, thus the two vectors are linearl independent However, v, v and v are linearl dependent, as with a correct scaling of v and v, we get c v v c v

2 Write the sstem x + + z = 4x z = 4 7x z = 9 in matrix form Solution : x = z If possible, compute the following matrix products (a) 4 (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) 7 5 d b c a ( )

3 Solutions: (a) (b) (c) not possible (d) (e) (g) (i) a b c d (f) (h) (j) (k) (l) ( 4 ) ( ad bc ) ad bc ( ) Introducing inverses for matrices (a) Find all vectors x such that A x = b, where A = b = 4 Solution : The matrix equation A x = b corresponds to the linear sstem x + = x + 4 = which has the unique solution x = 5/ (b) Prove : The matrix A = is invertible if and onl if ad bc (Hint : Consider the cases a = and a separatel) Solution : The matrix equation u v = x ields the (simultaneous) linear sstems (*) au + bx = av + b = cu + dx = cv + d = First assume that a = Note that the sstem on the left is inconsistent if b = Hence assume b This forces, in the sstem on the right, =, in the sstem on the left, x = /b

4 We are left with cu = d/b and cv = Again, if c ( =, the ) whole sstem is inconsistent Hence, assume c and the matrix admits inverse d/bc /c if and onl if bc /b Next, assume a Then one obtains from the linear sstem on the left (cf (*)) x(ad bc) = c and from the linear sstem on the right, (ad bc) = a In particular, the total sstem is inconsistent if ad bc = and admits a unique solution otherwise (c) Prove : If A is invertible, then its inverse is given b d b = ad bc c a Solution : Observe that ad bc is a necessar condition for the righthand side in the formula above to exist B definition A is invertible if there exists a matrix B such that AB = BA = You can check that the matrix A described b the formula satisfies these two equations (d) Use the formula in (c) to compute x in (a) Solution : Since 4 6, the matrix A is invertible and x = A b = 4 6 ( /) = (/) 5 6 The goal of this exercise is to give a geometric interpretation of the linear transformations defined b the matrices A = B = C = Start b showing the effect of these transformations on the letter L : x In each case, decide whether the transformation is invertible Find the inverse if it exists, and interpret it geometricall x x x x x Solutions : You can check that A =, B =, C = x 4

5 The matrix A scales vectors b a multiple of You can see that the letter L, once we appl A to it, is still sitting at( the ) origin but is now three times "bigger"; going on the vertical axis up to and on the horizontal axis to The inverse transformation would be a rescaling of / In fact, using the formula ( from ) Exercise 5(c), ou can check that the inverse matrix is given b / / The matrix B projects points onto the horizontal axis The letter L is reduced, under this projection, to the segment [, ] on the x-axis Intuitivel, there should be no well-defined inverse, since an point on the vertical line passing b (x, ) is a potential pre-image of the transformation Appling Exercise 5(b), ou can in fact check that the criterion for inverses is not fulfilled b the matrix B To understand the geometric action of the matrix C, it me easier to look at its effect on the letter L : x The letter L has been rotated ( b 9 o clockwise ) The inverse is the rotation of 9 o counterclockwise, given b 5