Linear Algebra Test Tomorrow

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Pavshaus wrote:

42 ... the answer is always 42.

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ampdecay wrote:

5675 character formula sheet in spoiler

Spoiler

|Theorem 2.1.1 If A is an nxn matrix, then regardless of what row or column of A is chosen, the number obtained by mulitplying the entries in that row or column by the corresponding cofactors and adding the resulting products is always the same.|Def.2 multiplying row or column by cofactors and adding is det(A) and the sums are cofactor exspansions of A:det(A)=a1jC1j+a2jC2j+...+anjCnj|Theorem 2.1.2 If A is an nxn triangular matrix then det(A) is the product of the diagonal entries|Theorem 2.2.1 Let A be a square matrix, if A has a row of zeros of a column of zeros then det(A)=0|Theorem 2.2.2 Let A be a square matrix, then det(A)=det(AT)|Theorem 2.2.3 Let A be an nxn matrix (a) If B is the matrix that results when a single row or single column of A is multiplied by a scalar k then det(B)=kdet(A) det(kA)=kndet(A).(b)If B is the matrix that results when two rows or two columns of A are interchanged then det(B)=-det(A). (c)If B is the matrix that results when a multiple of one row of A is added to another row (or columns) then det(B)=det(A)|Theorem 2.2.4 Let E be an nxn elementary matrix (a)If E results form multiplying a by nonzero number k det(E)=k. (B)If E results from interchanging two rows then det(E)=-1.(c)If E results from adding a multiple of one row to another det(E)=1.|Theorem 2.2.5 If A is a square matrix with two proportional rows(or columns) then det(A)=0|Theorem 2.3.1 If A,B, and C differ by only one row(or column) then det(C)=det(A)+det(B) that is, add differing rows of A and B to get the corresponding row(or column) of C|Theorem 2.3.3 A square matrix is invertible if and only if det(A)≠0|Theorem 2.3.4 If A and B are the same size square matrices det(AB)=det(A)det(B)|Theorem 2.3.5 If A is invertible det(A)-1=1/det(A)|Definition 2.3.1 adj(A) is the transpose of the cofactor matrix of A|Theorem 2.3.6 A-1=(1/det(A))adj(A)|Cramers Rule if Ax=B is a system of n linear equations in n unknowns where det(A)≠0 then the system has a unique solution. x1=det(A1)/det(A)...xn=det(An)/det(A) where A1 to An are the matrices formed when each column is replaced by the matrix B from A1 to An.|Theorem 2.3.8 If A is nxn matrix then the following statements are equivalent (a)A is invertible (b)Ax=0 has only the trivial solution (c) Reduced row echelon form of A is In (d) A can be expressed as a product of elementary matrices. (e) Ax=B is consistent for every nx1 matrix B (f) Ax=B has exactly one solution for every nx1 matrix B (g)det(A)≠0|Definition 3.1 all ordered n-tuples is called n-space denoted Rn|Definition 3.2 vectors v and w are equal when v1 = w1 ...vn = wn denoted v = w|Definition 3.3 v+w=(v1+w1,...vn+wn) kv=(kv1....kvn) -v=(-v1...-vn) w-v=w+(-v)|Theorem 3.3.1 u,v vectors k,m scalars(a) u+v=v+u (b)(u+v)+w=u+(v+w) (c)u+0=0+u=u (d)u+(-u)=0 (e)k(u+v)=ku+kv (f)(k+m)u=ku+km (g)k(mu)=(km)u (h)1u=u|Theorem 3.1.2 (a)0v=0 (b)k0=0 (c)-1v=-v|
Definitoin 4 w is linear combination of vectors when w=k1v1+...+knVn k are scalars called coeffcients|Defintion 3.2.1 the norm of a vector is ||v||=sqrt(v21 + … v2n)|Theorem 3.2.1 ||v||>=0 ||v||=0 if v=0 ||kv||=|k|||v||.|Definition 2 distance is d(u,v)=||u-v||.|Definition 3 dot product(inner product) is u•v=||u||||v||cosθ if u=0 or v=0 u•v=0|Definition 4 inner product u•v=u1v1+...unvn|Theorem 3.2.2 (a)u•v=v•u [symmetry] (b) u•(v+w)=u•v+u•w [distribution] (c)k(u•v)=(ku)•v [homgenenity] (d) v•v>=0 and v•v=0 iff v=0 [positivity]|Theorem 3.2.3 (a)0•v=v•0=0 (b) (u+v)•w=u•w+v•w (c)u•(v-w)=u•v-u•w (d)(u-v)•w=u•w-v•w (e)k(u•v)=u•(kv)|Theorem 3.2.4 Cauchy-Schwarz Inequality |u•v|<=||u||||v||.|Theorem 3.2.5 Triangle Inequality of Vectors ||u+v||<=||u||+||v||.|Defintion 3.3.1 u and v are orthogonal if u•v=0 also zero vector is orthogonal to all vectors and an orthogonal set of vectors means all pairs of vectors are orthogonal and is called an orthonormal set.|Theorem 3.3.1 ax+by+c=0 represents a line with normal n=(a,b) ax+by+cz+d=0 represents a plane with a normal n=(a,b,c)|projau=(u•a/||a||2)a|Theorem 3.3.4 The distance D between the point P0(x0,y0) and the line ax+by+c=0 is D=|ax0+byo+c|/sqrt(a2+b2) and distance D between the point P0(x0,y0,z0) and the plane ax+by+cz+d=0 is D=|ax0+byo+cz0+d|/sqrt(a2+b2+c2)|Theorem 3.4.1 the equation of the line through x0 that is parallel to v is x=x0+tv if x0=0 then the line passes through the origin and has the form x=tv|Theorem 3.4.2 the equation of the plane through x0 that is parallel to v1 and v2 is x=x0+t1v1+t2v2 if x0=0 then the plane passes through the origin and has the form x= t1v1+t2v2|Definition 3.4.3 If x0 and x1 are vectors in Rn then the equation x=x0+t(x1-x0) (0<=t<=1) defines the line segment from x0 to x1|Theorem 3.4.4 The general solution of a consistent linear system Ax=B can beobtained by adding any specific solution of Ax=B to the general solution of Ax=0|Definition 3.5.1 cross product u x v = (u2v3-u3v2, u3v1-u1v3, u1v2-u2v1)|Theorem 3.5.1 u x v is orthogonal to u and v|Theorem 3.5.3 ||u x v|| is the area of a parallelogram in 3-space.|Definition 4.1.1 Let V be an arbitary non empty set of objects on which two operations are defined: addition and multiplication by scalars. If the following axioms are satisified by all objects u,v and w in V and all scalars k and m, then we call V a vector space and we call the objects in V vectors. 1.If u and v are objects in V then u + v is in V 2. u+v=v+u 3. u+(v+w)=(u+v)+w 4. There is an object 0 in V called a zero vector for V such that 0+u=u+0=u 5. For each u in V there is an object -u in V called a negative of u such that u+(-u)=(-u)+u=0 6. if k is any scalar and u is any object in V then ku is in V 7.k(u+v)=ku+kv 8. (k+m)u=ku+mu 9. k(mu)=(km)u 10. 1u=u

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Asphael wrote:

Good luck. Also it seems like this is only Lineare Algebra I. LA II is a real bitch I can tell you that.

Btw I hope you can give proof for all the neat Theorems you got there. ;D

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ampdecay wrote:

At least I'm pretty sure I got an A this time around.