Section 1.6: R^{n} Thm (Cauchy-Schwarz Inequality): Let u and v ∈ R^{n}. Then

u · v ≤ ||u|| ||v||

Recall in R^{2} and in R^{3}, u · v = ||u|| ||v||cosθ Thm (Triangle Inequality): Let u and v ∈ R^{n}. Then

||u + v|| ≤ ||u|| + ||v||

A matrix is a vector of vectors Example of a 2 × 3 matrix:

[ \begin{matrix}1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} ]  [ \begin{matrix}1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} ]
[ \begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix} ]  [ \begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix} ]

Operations on Matrices A = (a_{ij}), B = (b_{ij}), C = (c_{ij}). Defn: Two matrices A and B are equal, if they have the same size and a_{ij} = b_{ij} for all i = 1, ..., n, j = 1, ..., m Defn: A + B = (a_{ij} + b_{ij}). Defn: cA = (ca_{ij}). Defn: −B = (−1)B. Defn: A − B = A + (−B). Defn: The zero matrix = 0 = (a_{ij}) where a_{ij} = 0 for all i, j. Defn: The identity matrix = I = (a_{ij}) where a_{ij} = 0 for all i ≠ j and a_{ii} = 1 for all i and I is a square matrix.

The transpose of the m × n matrix A = A^{T} = (a_{ji}).

Suppose A is an n × r matrix, B is an r × n. AB = C where

c_{ij} = row(i) of A · column(j) of B
= a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{ir}b_{rj}.

Ex: [ \begin{matrix}1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} ] [ \begin{matrix}2 & 3 \\ 10 & 1 \\ 0 & 4 \end{matrix} ] =

jth column of AB = A[jth column of B] In other words AB = A[b_{1}...b_{n}] = [Ab_{1}...Ab_{n}]

ith row of AB = [ith row of A]B

In other words AB = [ \begin{matrix}a(1) \\ . \\ . \\ . \\ a(n) \end{matrix} ] B = [ \begin{matrix}a(1)B \\ . \\ . \\ . \\ a(n)B \end{matrix} ]

Thm (Properties of matrix arithmetic) Let A, B, C be matrices. Let a, b be scalars. Assuming that the following operations are defined, then a.) A + B = B + A b.) A + (B + C) = (A + B) + C c.) A + 0 = A d.) A + (−A) = 0 e.) A(BC) = (AB)C f.) AI = A, IB = B g.) A(B + C) = AB + AC, . (B + C)A = BA + CA h.) a(B + C) = aB + aC i.) (a + b)C = aC + bC j.) (ab)C = a(bC) k.) a(AB) = (aA)B = A(aB) l.) 1A = A Defn.) −A = −1A Cor.) A0 = 0, 0B = 0 Cor.) a0 = 0 a.)(A^{T})^{T} = A b.) (A + B)^{T} = A^{T} + B^{T} c.) (kA)^{T} = kA^{T} d.) (AB)^{T} = B^{T} A^{T} Note

AB ≠ BA

It is also possible that AB = AC, but B ≠ C.

In particular it is possible for AB = 0, but A ≠ 0 AND B ≠ 0

Defn: A is invertible if there exists a matrix B such that AB = BA = I, and B is called the inverse of A. If the inverse of A does not exist, then A is said to be singular.

Ex: [ \begin{matrix}1 & 3 \\ 2 & 7 \end{matrix} ] [ \begin{matrix}7 & −3 \\ −2 & 1 \end{matrix} ] =

Note that if A is invertible, then A is a square matrix.

Thm: If A is invertible, then its inverse is unique.

Proof: Suppose AB = I and CA = I. Then, B = IB = CAB = CI = C.

Defn: A linear transformation is a function T : R^{n} → R^{m} that satisfies the following two conditions. For each u and v in R^{n} and scalar a,

i.) T (au) = aT (u)

ii.) T (u + v) = T (u) + T (v)

Thm: Let T : V → W be a linear transformation. Then T (0) = 0

Pf: T (0) = T (0 + 0) = T (0) + T (0)

Thm: Let A be an m × n matrix. Then the function

\begin{matrix}T : R^{n} → R^{m} \\ T (x) = Ax \end{matrix}

is a linear transformation.

Prove that T : R^{3} → R^{2}, T (x, y, z) = (4x − 5y, 8) is NOT a linear transformation.

Examples of linear transformations: Ex: If T : R^{2} → R is a linear transformation and T (1, 0) = 3, T (0, 1) = 4, then

T (x, y) = xT (1, 0) + yT (0, 1) = 3x + 4y = (3, 4) ( \begin{matrix}x \\ y \end{matrix} )

Thm 4.3.3: If T : R^{n} → R^{m} is a linear transformation, then T (x) = Ax where A = [T (e_{1})...T (e_{n})].

Proof: Let x = [ \begin{matrix}x_{1} \\ . \\ . \\ . \\ x_{n} \end{matrix} ] = x_{1}e_{1} + ... + x_{n}e_{n}.

Thus T (x) = T (x_{1}e_{1} + ... + x_{n}e_{n})

= x_{1}T (e_{1}) + ... + x_{n}T (e_{n})

= [T (e_{1})...T (e_{n})] [ \begin{matrix}x_{1} \\ . \\ . \\ . \\ x_{n} \end{matrix} ]