MA 106: Linear Algebra Instructors: Prof. Akhil Ranjan and Prof. S. - - PowerPoint PPT Presentation
MA 106: Linear Algebra Instructors: Prof. Akhil Ranjan and Prof. S. Krishnan January 4, 2017 1 / 42 Text and references Main Text: E. Kreyszig, Advanced Engineering Mathematics, 8 th ed. (Chapters 6 and 7) Additional references: 1) Notes by
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Main Text:
7) Additional references: 1) Notes by Prof. I.K. Rana 2) S. Kumaresan, Linear Algebra- A geometric approach. 3) Wylie and Barrett, Advanced Engineering Mathematics, 6th ed. (Chapter 13)
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Syllabus: see Math Dept webpage. Grading policy: Study hours:
1 Lectures - 21 hours? 2 Tutorials - 7 hours 3 Independent study - 42 hours January 4, 2017 3 / 42
1 Matrices 2 Addition, multiplication, transposition 3 Linear transformations and matrices 4 Linear equations and Gauss’ elimination 5 Row echelon forms and elementary row matrices 6 Reduced REF 7 Gauss-Jordan method for finding inverse January 4, 2017 4 / 42
Definition 1
A rectangular array of numbers,real or complex, is called a matrix. An array could be of any type of non mathematical objects too. Or more sophisticated mathematical objects like functions instead of numbers. E.g. sin x − cos x cos x sin x
Most of the topics today (sec 6.1 and 6.2) will be briefly reviewed and the details will be left for self study
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A = [ajk], 1 ≤ j ≤ m, 1 ≤ k ≤ n denotes an m × n matrix whose entry in jth row and the kth column is the number ajk. (Equivalently, kth entry in the jth row or equivalently jth · · · .) A = a11 · · · a1k · · · a1n . . . . . . . . . aj1 · · · ajk · · · ajn . . . . . . . . . am1 · · · amk · · · amn
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Some special cases are sometimes named differently
Definition 2
An m × 1 matrix is referred to as a column vector while a 1 × n matrix is referred to as a row vector . Examples :
1
1 −1 is a column.
2 [0 1 − 1 3 0] is a row.
Two matrices are said to be equal if and only if their corresponding entries are same.
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Definition 3
A matrix B is a transpose of A if the rows of B are the columns of A and vice versa. Thus, is A = [ajk] is an m × n then B is n × m matrix [brs] where brs = asr; 1 ≤ r ≤ n, 1 ≤ s ≤ m. The transpose of A is unique and is denoted by AT. Example: A = 5 −8 1 4
⇒ AT = 5 4 −8 1 . Exercise: Show that (AT)T = A.
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Definition 4
A matrix A is called symmetric (resp. skew-symmetric) if A = AT (resp. A = −AT). These are necessarily square matrices i.e.
Let A, B be real (or complex) matrices and λ ∈ R (or C) be a scalar.
Definition 5
(Addition) If A = [ajk] and B = [bjk] have the same order m × n, we define their addition to be A + B = [cjk] = [ajk + bjk].
Definition 6
(Scalar multiplication) The scalar multiplication of λ with A is defined as λA = [λajk].
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If A = [ajk] is m × n and B = [bkℓ] is n × p, then the product C := AB is a well defined m × p matrix cooked by the following recipe (called row by column multiplication): C = [cjℓ] where cjℓ =
n
ajkbkℓ; 1 ≤ j ≤ m, 1 ≤ ℓ ≤ p. aj1 · · · ajk · · · ajn b1ℓ . . . bkℓ . . . bnℓ
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Theorem 7 (Associativity)
If A, B, C are real (or complex) matrices such that A is m × n, B is n × p and C is p × q, then the products AB and BC are defined and in turn the products A(BC) and (AB)C are also defined and the latter two are equal. In other words: A(BC) = (AB)C Proof: Exercise.
Theorem 8 (Distributivity)
If B, C are real (or complex) m × n matrices, then A(B + C) = AB + AC if A is p × m. (B + C)A = BA + CA if A is n × q. Proof: Exercise.
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Theorem 9
Let A be m × n and B be n × p, then AB and BTAT are well defined and in fact (AB)T = BTAT. Proof: Omitted. Exercise: Let A = 4 9 2 1 6 , B = 3 7 2 8
BTAT and ATBT to verify the claim.
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Definition 10
Let v = v1 . . . vn and w = w1 . . . wn be column vectors of the same size n. Their dot product (or inner product or scalar product) is defined as v · w =
n
vjwj. It is interesting to observe that v · w = vTw as a 1 × 1 matrix, which being symmetric also equals wTv = w · v. Question: What about vwT? Is it defined? Is it also the dot product? [1.0]
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From now on the elements of Rn will be written as the column vectors of length n.
Definition 11
A map f : Rn − → R is called linear if it is of the form f (x) = a1x1 + a2x2 · · · + anxn for suitable constants a1, a2, ..., an. Here x1, x2, ..., xn are the entries of x usually called variables. If we view A = [a1 a2 · · · an] as a row vector, then f (x) = Ax in terms of matrix multiplication. More generally, an m × n matrix A can be viewed as a linear map Rn − → Rm via x → Ax. This viewpoint allows us to study matrices geometrically.
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Formal definition of a linear map (also called transformation)
Definition 12
A map f : Rn − → Rm is called linear if it satisfies (i) f(x + y) = f(x) + f(y) and (ii) f(λx) = λf(x) Remark: For an m × n matrix A, f(x) = Ax is an example of a linear map. These are no other examples of linear maps. A is called the matrix associated to the linear transformation f. Example 1: Show that the range of 1 −1
→ R2 is a line through 0. A(t) = 1 −1
t −t
x(t) y(t)
are parametric equations of the line x + y = 0 through 0.
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Example 2:Determine domain and range and also show that 1 −1 −1 2 1 has a plane through 0 as its range. (i) The matrix is 3 × 2, so that’s a linear transformation R2 − → R3. (ii) B u v
1 −1 −1 2 1 u v
u − v −u + 2v v = x(u, v) y(u, v) z(u, v) , say.
plane through 0.
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Example 3:Consider the matrix: 1 1 1
(i) Unit square {0 ≤ x ≤ 1, 0 ≤ y ≤ 1}, (ii) Unit circle {x2 + y2 = 1} and (iii) Unit disc {x2 + y2 ≤ 1}. (i) Unit square {0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has vertices
1
1 1
1
1
2 1
1 1
with these vertices. (ii) A x y
x + y y
u v
⇒ x y
u − v v
x2 + y2 = 1 = ⇒ u2 − 2uv + 2v2 = 1 which is an ellipse. (iii) Elliptic disc u2 − 2uv + 2v2 ≤ 1 enclosed by the ellipse u2 − 2uv + 2v2 = 1.
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Example 4: Consider the matrix: 1 1 1 1
(i) Unit square {0 ≤ x ≤ 1, 0 ≤ y ≤ 1}, (ii) Unit circle {x2 + y2 = 1} and (iii) Unit disc {x2 + y2 ≤ 1}. (i) Unit square {0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has vertices
1
1 1
1
1 1
2 2
1 1
segment
2 2
(ii) A x y
x + y x + y
u v
x2 + y2 = 1 = ⇒ min (x + y) = − √ 2, max (x + y) = √ 2 = ⇒ a line segment from − √ 2 √ 2
√ 2 √ 2
(iii) the line segment above in (ii). [1.5]
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