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 + v2 = u2 + v2
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 nonzero 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 nonzero entry of every nonzero 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(GaussJordan 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 nonzero 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 GaussJordan 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 nonzero 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 GramSchmidt process)
The GramSchmidt 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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 Fall '08
 KLEMES
 Math, Linear Algebra, Dot Product, UK, Det, elementary row operations

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