3.4
Linear Dependence and Span
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Linear Combination
Denition 1 Given a set of vectors cfw_v1 , v2 , . . . , vk in a vector space V , any vector of the form
v = a1 v1 + a2 v2 + . . . + ak vk
for some scalars a1 , a2 , . . . , ak , is called a
4.1, 4.2
Determinants
P. Danziger
1
Determinants
Every n n matrix A has an associated scalar value called the determinant of A, denoted by det(A)
or |A|.
The determinant gives the (hyper)volume of the unit (hyper)cube after it has been transformed by
A.
N
4.3
Cross Product
P. Danziger
1
Cross Product
Big Note Cross product works in R3 ONLY
1.1
Denition
Given two non parallel vectors in R3 , u and v, there is a vector w which is perpendicular to both
of them. The direction of this vector is unique, up to or
Basis and Dimension
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Basis and Dimension
Denition 1 A basis of a vector space V , is a set of vectors B = cfw_v1 , v2 , . . . , vn such that
1. cfw_v1 , v2 , . . . , vn span V,
2. cfw_v1 , v2 , . . . , vn are linearly independent and hence
4.3
Cramers Rule
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Cramers Rule
Cramers rule is a method for solving n n systems of equations using determinants. Generally it
is less preferable than Gaussian elimination or Gauss-Jordan as there are more operations involved.
However, in some
7.11, 8.1, 8.2
Diagonalization
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Change of Basis
Given a basis of Rn , B = cfw_v1 , . . . , vn , we have seen that the matrix whose columns consist of these
vectors can be thought of as a change of basis matrix.
AB =
v1 v2 . . . vn
Given a vec
3.4
Row Space, Column Space and Nullspace
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Nullspace
Denition 1 Given an m n matrix A The Nullspace of A 1s the set of solution to the equation
Ax = 0.
Notes
The Nullspace of A = ker(A).
When we are asked to give a subspace (such as the nul
3.6
Special Matrices
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Triangular Matrices
Denition 1 Given an n n matrix A
A is called upper triangular if all entries below the main diagonal are 0.
A is called lower triangular if all entries above the main diagonal are 0.
A is called di
6.1 - 6.4
Properties of Transformations
P. Danziger
Transformations from Rn Rm
1
1.1
General Transformations
A general transformation maps vectors in Rn to vectors in Rm . We write T : Rn Rm to indicate
this.
Example 1
1. Given T : R2 R2 , T (x, y) = (x +
7.9
Gram-Schmidt Process
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Orthonormal Vectors and Bases
Denition 1 A set of vectors cfw_vi | 1 i n is orthogonal if vi vj = 0 whenever i = j and
1 i=j
orthonormal if vi vj =
0 i=j
For ease of notation, we dene the the Kronecker delta function
4.3
Adjoints
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The Adjoint & Inverses
Denition 1 Given an n n matrix A, the adjoint of A is the transpose of the matrix of cofactors.
A11 A12
A21 A22
.
.
.
.
.
.
An1 An2
. . . A1n
. . . A2n
.
.
.
.
. . . Ann
T
=
A11 A21
A12 A22
.
.
.
.
.
.
A1n
7.1
Eigenvalues & Eigenvectors
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Eigenvalues
Denition 1 Given an n n matrix A, a scalar C, and a non zero vector v Rn we say that
is an eigenvalue of A, with corresponding eigenvalue v if
Av = v
Notes
Eigenvalues and eigenvectors are only de