Linear Algebra 1 Joseph Spring School of Computer Science 3COM0164 Quantum Computing / QIP Lecture - Linear Spaces 1 1
Areas for Discussion Linear Space / Vector Space Definitions and Fundamental Operations Spanning Sets, Bases and Linear Independence Linear Operators and Matrices Definitions Identity Operator Zero Operator Operator Composition Matrix Representations Examples The Pauli Matrices Lecture - Linear Spaces 1 2
Linear Space / Vector Space Lecture - Linear Spaces 1 3
Fields In the following definition for a linear space we refer to a field as being composed of a set of scalars In general the fields we work with will be complex numbers and sometimes real numbers So as to maintain a sense of generality we present and define a field as follows: Lecture - Linear Spaces 1 4
Fields Let F denote the field 1.Addition α, β, χ F, α+ β F, the sum of α and β s.t. α + β = β + α addition is commutative (α + β) + χ = α + (β + χ) addition is associative! Scalar 0 (called zero) s.t. α F, α + 0 = α α F,! Scalar - α F s.t. α + (- α) = 0 Lecture - Linear Spaces 1 5
Fields 2. Multiplication α, β, χ F, αβ F, the product of α and β s.t. αβ = βα multiplication is commutative (αβ)χ = α(βχ) multiplication is associative! non zero scalar 1 F (called one) s.t. α F, α1 = α α F, α 0 ;! scalar α -1 F s.t. αα -1 = 1 3. Multiplication is distributive w.r.t. addition α, β, χ F α(β + χ) = αβ + αχ Lecture - Linear Spaces 1 6
Fields - Examples 1. (, +, ) 2. (, +, ) 3. (, +, ) Exercise 1. Show that 1 and 2 above are fields. We will return to 3. Lecture - Linear Spaces 1 7
Linear Space - Definition A Linear (Vector) Space consists of a set V of objects called vectors (denoted by v 1 >, v 2 >,, v n >) and a field F whose elements, (denoted by λ, µ, ν, ) are referred to as scalars s.t. 1.Addition Laws v 1 > + v 2 > = v 2 > + v 1 > Commutative Law ( v 1 > + v 2 >) + v 3 > = v 1 > + ( v 2 > + v 3 >) Associative Law v 1 > + 0 = 0 + v 1 > = v 1 > Additive Identity v 1 > + ( - v 1 > ) = 0 Additive Inverse Lecture - Linear Spaces 1 8
Linear Space - Definition 2. Multiplication α, β F, v 1 >, v 2 > V, α v 1 > V, the product of α and v 1 > s.t. α(β v 1 >) = (αβ) v 1 > 1 v 1 > = v 1 > α ( v 1 > + v 2 > ) = α v 1 > + α v 2 > multiplication by scalars is distributive w.r.t. vector addition (α + β) v 1 > = α v 1 > + β v 1 > multiplication by vectors is distributive w.r.t. scalar addition Lecture - Linear Spaces 1 9
Linear Space - Comments The above definition is considered a convenient characterisation of the basic objects that we wish to study The relation between a linear space V and its underlying field F is usually made clear by referring to V as a linear (vector) space over F We note that we have employed the Quantum Mechanical notation used in physics for a vector v 1 > This is sometimes referred to as a ket A linear (vector) subspace W of a linear (vector) space V is also a vector space Lecture - Linear Spaces 1 10
Linear Spaces - Examples 1. The set of directed line segments in the Cartesian plane forms a linear (vector) space if we define v 1 > + v 2 > to be appropriate addition and define α v 1 > to be appropriate multiplication. What were these appropriate relations? 2. The set of polynomials with real coefficients with usual algebraic addition and multiplication Lecture - Linear Spaces 1 11
Linear Spaces - Examples n 3. Let denote the set of all n tuples of real numbers ( n = 1, 2, 3, ) with z 1 = (x 1, x n ) and z 2 = (y 1, y n ). Let z 1 + z 2 = ( x 1 + y 1,, x n + y n ) α z = 1 ( α x 1,, α x n ) n 0 = ( 0,, 0 ) - z 1 = ( - x 1,, - x n ) n Then is a real linear space: the n dimensional real coordinate space Lecture - Linear Spaces 1 12
Linear Spaces - Examples 4. The most trivial linear space is the space containing one element, the zero vector Exercise 2. Show that the above are linear spaces Lecture - Linear Spaces 1 13
Spanning Sets Definition A spanning set for a vector space is a set of vectors { v 1 >,, v n > } each from V s.t. v > V, {α 1,, α n } each from F, s.t. v > can be written as a linear combination n v > = α i v i > i= 1 Lecture - Linear Spaces 1 14
Spanning Sets Example 2 A spanning set for the vector space is the set 1 v 1 > =, and v 2 > = 0 0 1 Since α α 1 2 v > = in can be written as 2 v > = α 1 v 1 > + α 2 v 2 > Lecture - Linear Spaces 1 15
Spanning Sets We say that the vectors v 1 > and v 2 > span the vector space We note that a vector space may in general have many different spanning sets e.g. 1 1 2 1 α α v 1 > =, and v 2 > = 1 Since v > = in can be written as a linear combination of v 1 > and v 2 > α 2 2 1 2 v > = v 1 > + v 2 > 2 1 2 + α α1 α2 2 2 1 1 Lecture - Linear Spaces 1 16
Linear Dependence and Independence A set of non-zero vectors { v 1 >,, v n > } is said to linearly dependent if a set of scalars {α 1,, α n } with α i 0 for at least one value of i s.t. α 1 v 1 > + α 2 v 2 > + α n v n > = 0 A set of vectors is said to be linearly independent if it is not linearly dependent Lecture - Linear Spaces 1 17
Bases and Dimension It can be shown that any two sets of linearly independent vectors which span a vector space V contain the same number of elements We define a basis for the vector space V to be any such linearly independent spanning set of V It can be shown that a basis always exists for a vector space V The number of elements in the basis is defined to be the dimension of the vector space V, denoted dim V We are only interested in finite dimensional vector spaces Lecture - Linear Spaces 1 18
Linear Dependence and Independence Question 4. Show that ( 1, -1 ), ( 1, 2 ) and ( 2,1 ) are linearly dependent Lecture - Linear Spaces 1 19
Linear Operators and Matrices Lecture - Linear Spaces 1 20
Linear Operators and Matrices Linear Operators and Matrices Definitions Identity Operator Zero Operator Operator Composition Matrix Representations Examples The Pauli Matrices Lecture - Linear Spaces 1 21
Linear Operators Definition A linear Operator between two linear (Vector) spaces V and W is defined to be a mapping A : V W which is linear in its inputs: A αi vi > = αia vi > i ( ) We usually write A v > for A( v > ) i Lecture - Linear Spaces 1 22
Identity Operator We often say that a linear operator A is defined on a vector space. By this we mean that A is a linear operator from V to V The identity operator I V is an important operator on any vector space V defined by I v > = v > V for all v >. We often drop the subscript V and write I if it is clear which vector space is under discussion Lecture - Linear Spaces 1 23
Zero Operator and Composition Another important operator is the zero operator which maps all vectors to the zero vector 0 v > = 0 0 Let X, Y and Z be linear (vector) spaces with A : X Y and B : Y Z be linear operators. Then the composition of B with A is denoted with BA : X Z and defined as BA( v > ) = B(A( v > )) We write BA v > as an abbreviation for BA( v > ) Lecture - Linear Spaces 1 24
Matrix Representations Lecture - Linear Spaces 1 25
Matrix Representations One of the most important topics involving linear operators acting on finite dimensional spaces is that of matrices acting on the same spaces It turns out that the linear operator and matrix viewpoints turn out to be equivalent If you are familiar with matrices then the linear operator viewpoint is an alternative way of discussing the same ideas So why not stick with matrices??? Well matrices don t extend very well to infinite dimensional spaces (/bases) which general quantum theory works With Quantum Computing we have finite dimensional bases Lecture - Linear Spaces 1 26
Matrix Representations Let A be an m n matrix with entries A ij Then: A n columns A 11 A 12... A 1n A 21 A 22... A 2n =............ A m1 A m 2... A mn m rows Lecture - Linear Spaces 1 27
Example Let A be a 2 3 matrix and B be a 3 2 matrix For example: Comments? A B 3 4 7 = 6 2 1 5 2 = 7 1 2 3 Lecture - Linear Spaces 1 28
Matrix Representations We note: 1. A : n I I where I is the field of the given linear/vector space m In general I is or 2. The composition AB and BA are both defined for the given matrices. In general this is not the case. Justify! Lecture - Linear Spaces 1 29
Matrix Representations Let A denote a matrix acting on some linear spaace α 1 α 2 Let ψ >=. denote a vector fom the n dimensional space. α n Then A is a linear operator since: n n n A1 jµα j µ A1 jα j A1 jα j j= 1 j= 1 j= 1 α 1 µα 1 α n n n 1 α 2 µα 2 A2 jµα j µ A2 jα j A2 j α j α 2 j= 1 j= 1 j A( µ ψ >= ) A( µ. ) = A(. ) = = = µ = 1.. µ A. ( )... = = µ A ψ >... α n µα. n n n α n n Amjµα j µ Amj αj Amjα j j= 1 j= 1 j= 1 Lecture - Linear Spaces 1 30
Matrix Representations of Linear Operator To express a linear operator A: V W finite dimensional spaces as a matrix we proceed as follows: Let { v i >} m i=1 be a basis for V, and { w i >}n i=1 be a basis for W Since the operator A sends each v i > in V into W constants A ij field I (real or complex for us) s.t. A v j > = A ij w i > The matrix with entries A ij is said to be the matrix representation of the linear operator A Lecture - Linear Spaces 1 31
Matrix Representations of Linear Operator Linear Operators A Linear Space V Linear Space W v i > A ij w i > Lecture - Linear Spaces 1 32
Pauli Operators Lecture - Linear Spaces 1 33
The Pauli Operators The following 2x2 matrices frequently occur in QIP/QC and are to be found in many of the Exercises in Nielson and Chuang 10 01 σ0 I, σ1 σx X 01 10 0 i 1 0 σ2 σy Y, σ3 σz Z i 0 0 1 Lecture - Linear Spaces 1 34
Summary Linear Space / Vector Space Definitions and Fundamental Operations Spanning Sets, Bases and Linear Independence Linear Operators and Matrices Definitions Identity Operator Zero Operator Operator Composition Matrix Representations Examples The Pauli Matrices Lecture - Linear Spaces 1 35