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Berkeley MATH 110 - Lecture Notes

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Math 110 - Fall 05 - Lectures notes # 24 - Oct 24 (Monday)Now we begin Chapter 4, determinants. Determinants are usefulin several fields of mathematics:Linear Algebra: deciding if A is invertible, defining eigenvaluesGeometry: finding volumes of parallelograms (in 2D) orparallelepipeds (in any dimension)Calculus: changing variables in a multiple integralThere are several equivalent definitions, useful in differentsituations, all derivable from one another. We start with n=1and n=2, for which some definition look way too complicated,and then see that for higher dimensions they work best.We will assume the field F does not have characteristic 2 when needed.The determinant of a 1x1 matrix A = [a] is just a.The determinant of a 2x2 matrix A = [ x1 y1 ] is[x2y2](1) Explicit formula: det(A) = x1*y2 - y1*x2(2) Recursive formula: det(A) = x1*det([y2]) - y1*det([x2])This is identical to the explicit formula in the 2x2 case,but will extend to larger n(3) "Oriented" area of a parallelogram: Let P be the parallelogram with3 corners at (0,0), (x1,y1) and (x2,y2).This means the 4th corner must be (x1+x2,y1+y2) (picture).Recall area of parallelogram = Base x heightASK&WAIT: Why?Consequence: can take one side, "slide" it parallel to otherside without changing area. For example, we could replace(x2,y2) by (x2,y2) - c*(x1,y1) for any c without changing area.Let’s pick c to make it easy to figure out base and height. (picture):First, "slide" top edge to put corner on y axis, i.e.pick c so (x2,y2) - c*(x1,y1) = (0,y’) for some y’. Thusc = x2/x1 (assume x1 nonzero for the moment) and y’ = y2-(x2/x1)*y1Second, slide right edge to put corner on x axis, i.e.pick c’ so (x1,y1) - c’*(0,y’) = (x1,0).We see we get a rectangle with the same area as the parallelogram:1area = base*height = x1 * ( y2 - (x2/x1)* y1) = x1*y2 - x2*y1We call this the "oriented area" because it could be negative.Its absolute value is the "usual area". If x1 is zero, we don’thave to do one of the "slides", and end up with the same answer.The orientation is easy to understand geometrically in this 2 by 2 case:If moving from side 1 to side 2 within the parallelogram means youmove counterclockwise, the orientation = 1 (positive), other it is -1.In high dimensions, it is harder to explain geometrically, which iswhy we use the other, algebraic definitions.(4) LU factorization: Assuming x1 is nonzero, we can do the LU factorizationA=[x1y1]=[1 0]*[x1 y1 ]=L*U[ x2 y2 ] [ x2/x1 1 ] [ 0 y2 - (x2/x1)*y1 ]and just take the product of the diagonal entries of U:x1 * (y2 - (x2/x1)*y1) = x1*y2 - x2*y1Note that the diagonal entries of U are the same numbers we getfrom "sliding" edges. This is not a coincidence.(We will later generalize this to the case A = P_L * L * U * P_R)(5) Axiomatic definition: The determinant of an n x n matrix is thefunction det: M_{n x n}(F) -> F satisfying(1) det(A) is a linear function of each row (or column).In other words, if A(x) is a matrix with row i equal to x(and the other rows fixed), thendet(A(c*x + y)) = c*det(A(x)) + det(A(y))(2) swapping two rows (or columns) of A changes the sign of det(A)(provides "orientation")(3) det(I) = 1. (obvious volume of unit "cube").To illustrate axiom (1) in the 2 x 2 case:det( [ c*x1 + d*x1’ c*y1 + d*y1’ ] ) =[x2 y2]= c * det([ x1 y1 ]) + d* det([ x1’ y1’ ])[x2y2] [x2 y2 ]So even though the determinant itself is a polynomial, it isactually a linear function of any row or column, which we willfind very useful.One might ask why "oriented" area instead of just area? I.e. why nottake absolute values in all these definitions? Because then we would2lose the linearity property just described, which we will need.So if you want the usual area (or volume...) just take the absolutevalue at the end.(6) Product of A’s eigenvalues: But we haven’t defined eigenvalues yet!Now let’s look at the definitions for n>2:(1) Explicit formula: Written out, it would be a polynomial of degree n,with n! terms. n! grows quickly: 3! = 6, 4! = 24, 5! = 120, 10! = 362880, ...so this is not so useful as later definitions.(2) Recursive formula: This is the starting definition used by the textbook:Def: Let A be an n by n matrix. Then A^tilde_ij is the n-1 by n-1matrix gotten by deleting row i and column j of A.Recursive definition of Determinant: If A = [a] is 1 by 1, det(A) = a.Otherwise,det(A) = sum_{j=1 to n} (-1)^(1+j) * A_1j * det(A^tilde_1j)= A_11*det(A^tilde_11) - A_12*det(A^tilde_12)+ A_13*det(A^tilde_13) - ...(3) Oriented volume of a parallelepiped: In the 3 by 3 case, think of theparallelepiped P with corner at the origin and the points defined by the 3 rowsof A. Altogether A has 8 corners, whose coordinates are gotten by taking summingall possible subset of the 2^3 = 8 rows of A. (picture).P’s volume (with an appropriate orientation or sign) is det(A).In the n by n case, P will also have corners at the origin and the n pointsdefined by the n rows of A. Altogether A has 2^n corners, gotten from summingall possible subsets of A’s rows. Again, P’s volume(with an appropriate orientation) is det(A). The easiest way to see thisis from the other definitions, and as in the 2 by 2 case interpreting themas changing the parallelepiped ("sliding" edges) to another one with thesame volume and all perpendicular edges (a "box") whose volume is justthe product of the edge lengths.(4) LU factorization. Using A = P_L * L * U * P_R will be the best way toactually compute det(A) in practice for large matrices:det(A) = { 0 if rank(A) < n{ det(P_L)*det(P_R)*U_11*U_22*...*U_nn if rank(A) = n3where det(P_L) and det(P_R) are both either +1 or -1, and easy to figure out.We will return to this once we understand the other definitions.(5) Axiomatic Definition: This is same as above: The determinant ofan n x n matrix is the function det: M_{n x n}(F) -> F satisfying(1) det(A) is a linear function of each row.(2) swapping two rows of A changes the sign of det(A).(3) det(I) = 1.Our next goal is to how that the recursive formula satisfies all theseproperties of the Axiomatic definition.Thm 1: det(A), as given by the recursive formula, is a linear function of eachrow. In other words if A is an n by n matrix, with its i-th row writtenas a = c*y + z, where y and z are vectors, and c is a scalar,then we can write det(A) = c*det(Y) + det(Z) whereY = A except Y’s i-th row is y, andZ=Aexcept Z’s i-th row is z.Proof: We use induction. In the 1 x 1 base case the result is immediate:det([a]) = a = c*y + z


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Berkeley MATH 110 - Lecture Notes

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