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Equation sheet

Equation sheet - MATH 133 Formula Sheet Deﬁnition(Norm of...

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Unformatted text preview: MATH 133 - Formula Sheet Deﬁnition(Norm of a vector) v1 v2 If v = . in Rn the norm of v is given by ||v|| = . . vn Deﬁnition(Dot Product) u1 u2 If u = . and v = . . un v1 v2 . . . vn 2 2 2 v1 + v2 + · · · + vn then the dot product of u and v is deﬁned by u · v = u1 v 1 + u2 v 2 + · · · + un v n If θ is the angle between u and v then u · v = ||u|| ||v|| cos θ The Following Are Equivalent 1. u is orthogonal to v 2. u · v = 0 3. ||u + v||2 = ||u||2 + ||v||2 Deﬁnition(Orthogonal projection) Let u = 0 and v be two vectors in Rn , the projection of v onto u is given by u·v u u·u proju v = Equation of a line in R2 If L is a line in R2 its general equation is given by ax + by = c where n = a b is a normal vector for L. Equation of a line in R3 In R3 the vector equation of a line is given by (L) : p + td a where p = (x0, y0, z0 ) is a point on the line and d = b is its directional vector. c x = x0 + at y = y0 + bt In parametric form we write (L) : z = z0 + ct 1 Equation of a plane in R3 a Let p = (x0, y0 , z0) be a point in the plane, n = b a vector normal to the c plane, u and v two vectors parallel to the plane (but not parallel to each other) and x = (x, y, z ) any point in the plane then: • Normal form: n · (x − p) • General form: ax + by + cz = d • Vector Form: x = p + su + tv Some distances • distance from a point B to a line (L) Get a point A on the line. Let d be the directional vector of the line. Denote the vector AB by v. dist(B, L) = ||v − projd v|| • distance from a point B (x0, y0) to a line (L) : ax + by = c (in R2) |ax0 + by0 − c| √ a 2 + b2 • distance from a point B (x0, y0, z0) to a plane (P ) : ax + by + cz = d dist(B, L) = |ax0 + by0 + cz0| dist(B, P ) = √ a 2 + b 2 + c2 Theorem (Number of solution of a system of linear equations ) A system of linear equations can have only one of the following • No solution (inconsistent system) • A unique solution (consistent system) • A inﬁnte number of solutions (consistent system) Deﬁnition(Elementary Row Operations, ERO) The three elementary row operations are: 1. Interchange two rows. 2. Multiply (or divide) a row by a non-zero constant. 3. Add a multiple of a row to another. Deﬁnition(Reduced Row Echelon Form, RREF) A matrix is in Reduced Row Echelon Form if it satisﬁes the following 4 conditions 2 1. All zero rows are at the bottom. 2. The ﬁrst non-zero entry of every non-zero row is a 1 (leading one). 3. Leading ones go from left to right. 4. All entries above and below any leading one are zero. If a matrix satisﬁes only the ﬁrst 3 conditions above then we say it is in Row Echelon Form (REF). Deﬁnition(Gauss-Jordan elimination process) This is the process of applying the ERO’s to a matrix to get it to RREF. Deﬁnition(Rank of a matrix) The rank of a matrix is the number of non-zero rows in its RREF or REF . Deﬁnition(Linear combination) A vector u is a linear combination of the vectors v1 , v2, . . . , vn if we can ﬁnd scalars a1, a2, . . . , an such that u = a1 v1 + a2 v2 + · · · + an vn Deﬁnition(Span, Spanning Set) Given a set S = {v1 , v2, . . . , vn } of vectors in Rn : • Span(S ) = the set of all linear combinations of the vectors in S . • If span(S )= Rn then we say S is a spanning set for Rn . Deﬁnition(Linear independance) A set v1, v2 , . . . , vn of vectors in Rn is said to be linearly independant if the only solution to the equation c1 v 1 + c2 v 2 + · · · + cn v n = 0 is c1 = c2 = · · · = cn = 0. Otherwise the vectors are called linearly dependant (which also means that at least one of them can be written as a linear combination of the others). Deﬁnition(Symmetric matrix) A square matrix is symmetric if A = AT . Deﬁnition(Inverse of a Square Matrix) Given a square matrix A its inverse (if it exists) is the matrix denoted by A−1 such that AA−1 = A−1 A = I . If the matrix is a 2 2 matrix we use the formula ab cd −1 = 1 ad − bc 3 d −b −c a provided that the determinant of A, det(A) = ad − bc = 0. For a matrix of higher dimensions the process looks like this: [A | I ] → Gauss Jordan Process → I | A−1 If the matrix is not invertible (i.e. does not have an inverse) we will not get the identity on the left side after applying the Gauss-Jordan process. Deﬁnition(Elementary Matrix) An elementary matrix is a matrix that can be obtained by applying one Elementary Row Operation to the identity matrix. Deﬁnition(Row Space, Column Space, Null Space) Let A be an m × n matrix, • The row space of A = span(Rows of A). • The Column space of A = span(Columns of A). • The Null space is the subspace of Rn spanned by the solutions of the homogeneous system Ax = 0. Deﬁnition(Basis) A basis of a subspace S of Rn is a set of vectors that span S and are linearly independant. Deﬁnition(Rank) The rank of a matrix A (denoted by rank(A))is the dimension of its row space (or column space since they’re equal) Deﬁnition(Nullity) The nullity of a matrix A (denoted by nullity(A)), is the dimension of its Null space. Theorem (The Rank Theorem ) For any Am×n , rank(A) + nullity(A) = n. Deﬁnition(Linear Transformation) A transformation T : Rn → Rm is called a linear transformation if it satisﬁes 1. T (u + v) = T (u) + T (v) 2. T (k u) = kT (u) We usually check if T is a linear transformation by checking that T (c 1 v 1 + c 2 v 2 ) = c 1 T (v 1 ) + c 2 T (v 2 ) for c1 , c2 scalars and v1 , v2 in Rn . 4 Deﬁnition(Minor) Given An×n , the minor of entry ij is denoted by Aij and is the determinant of the matrix we get from A by removing row i and column j . Deﬁnition(Cofactor) Cij = (−1)i+j Aij Deﬁnition(Determinant of an n × n matrix) Given an n × n matrix A (n 2) det(A) = ai1Ci1 + ai2Ci2 + · · · + ain Cin by expanding along the ith row. det(A) = a1j C1j + a2j C2j + · · · + anj Cnj by expanding along the j th column. Properties of the determinant function Given an n × n matrix A • If A has a zero row or zero column then det(A) = 0. • If we get matrix B by interchanging two rows of A then det(B ) = − det(A). • If we get matrix B by multipying one row of A by k = 0 then det(B ) = k det(A). • If we get matrix B by adding a multiple of a row to another of matrix A then det(B ) = det(A). • det(kA) = k n det(A). • det(AT ) = det(A). • det(AB ) = det(A) det(B ) 1 . • det(A−1 ) = det(A) Deﬁnition(Eigenvalue, Eigenvector, Eigenspace) Given An×n a scalar λ is an eigenvalue of A if there is a non-zero vector x such that Ax = λx. The eigenvalues of A are the roots of the characteristic polynomial given by det(A − λI ); (we solve det(A − λI ) = 0). In this case x is called an eigenvector or A corresponding to λ. The collection of all eigenvectors corresponding to λ along with the zero vector form the eigenspace of λ denoted by Eλ . Deﬁnition(Similar Matrices) Given A and B two n × n matrices. A is said to be similar to B (written A ∼ B ) if there is an invertible matrix P such that P −1 AP = B . 5 Deﬁnition(Diagonalizable matrix) An n × n matrix A is diagonalizable if there is a diagonal matrix D that is similar to A. i.e. If there is a diagonal matrix D and an invertible matrix P such that D = P −1 AP . Theorem (when is a matrix diagonalizable? ) An n × n matrix A is diagonalizable if one of the following is true • A has n distinct eigenvalues. • For each eigenvalue the geometric multiplicity is equal to the algebraic multiplicity. Deﬁnition(Orthogonal set) A set of vectors {v1, v2, . . . , vn } is an orthogonal set if any two vectors in the set are orthogonal. (i.e. vi · vj = 0 for all i, j = 1, . . . n). Deﬁnition(Orthogonal basis) An orthogonal basis is a basis that is also an orthogonal set. Deﬁnition(Orthogonal matrix) An m × n matrix Q is called orthogonal if QT Q = In . (The columns of Q form an orthonormal set) Theorem (Important property about Orthogonal matrices ) If Q is a square orthogonal matrix then QT = Q−1 . Deﬁnition(Orthogonal complement) Let W be a subspace of Rn . We say that a vector v in Rn is orthogonal to W if v is orthogonal to every vector in W . The set of all vectors that are orthogonal to W is called the Orthogonal complement of W and denoted by W⊥ . Theorem (Important theorem to ﬁnd W⊥ ) If A is an m × n matrix then then (row(A))⊥ = null(A) and (col(A))⊥ = null(AT ) Deﬁnition(Orthogonal projection of v onto W ) Let W be a subspace of Rn and let {u1 , u2, . . . , uk } be an orthogonal basis for W . For any vector v in Rn , the orthogonal projection of v onto W is given by projW v = u1 · v uk · v u1 + · · · + uk u1 · u1 uk · uk Deﬁnition(The Gram-Schmidt process) The Gram-Schmidt process is the process we use to transform a basis into an orthogonal basis. It works as follows: Given {x1, x2, . . . , xk } a basis for a subspace W of Rn 6 v1 = x1 v2 = x2 − projv1 x2 v3 = x3 − projv1 x3 − projv2 x3 . . . vk = xk − projv1 xk − projv2 xk − · · · − projvk xk Finally we have {v1, v2, . . . , vk } an orthogonal basis for W . 7 ...
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