Chapter 3

Question 1

Def: matrix

Answer 1

mxn matrix A is a rectangular array with m rows and n columns
Entry in ith row and jth column by aij

Two mxn matrices A and B are equal aij=bij for all 1

Question 2

Def: matrix addition and scalar multiplication

Answer 2

Addition:
(A+B)ij=(A)ij+(B)ij

Scalar multiplication:
(cA)ij=c(A)ij

Question 3

Theorem 3.1.1

Answer 3

Mmxn(R) is a vector space with 0=0m,n = [mxn matrix with all 0 entries]

***matrix scalar multiplication and addition satisfy same 10 axioms as ch. 1

Question 4

Def: transpose

Answer 4

Given AEMmxn(R), the transpose of A is A^TEMmxn(R) given by (A^T)ij=(A)ji

Question 5

Theorem 3.1.2 (properties of transpose) ««««

Answer 5

1) (A^T)^T = A
2) (A+B)^T = A^T + B^T
3) (cA)^T = cA^T

Question 6

Def: matrix-vector multiplication

Answer 6

Ax = [v1…vn] [x1…xn]T, proceed as normal

Question 7

Theorem 3.1.4 (properties of matrix vector multiplication) ><><><><><><><><><><><><><><>

Answer 7

1) A(x+y) = Ax + Ay
2) c(Ax) = (cA)x = A(cx)
3) (Ax)^T=x^TA^T

Question 8

Def: matrix multiplication

Answer 8

For an mxn matrix A ans an nxp matrix B

AB = A[b1…bp] = [Ab1…Abp] EMmxp

(AB)ij = Σ(A)ik(B)kj=[dot product of the ith row of A and the jth column of B]
(Roman Catholic)

Question 9

Theorem 3.1.5 (properties of matrix multiplication)

Answer 9

1) A(B+C) = AB + AC
2) t(AB) = (tA)B = A(tB)
3) A(BC) = (AB)C
4) (AB)^T = B^TA^T (not commutative!)

Question 10

Def: identity matrix

Answer 10

The nxn identity matrix, denoted by I or In, is the matrix such that (I)ii = 1 for 1

Question 11

Theorem 3.1.7

Answer 11

AI = A = IA

Question 12

Def: block matrix

Answer 12

If A is an mxn matrix, then we can write A as the kxl block matrix
A = [A11 … A1l]
[… … …] (this is one matrix)
[Ak1 … Akl]
Aij is a block such that all blocks in the ith row have the same number of rows and all blocks in the jth column have the same number of columns

Question 13

Def: linear mapping/linear operator

Answer 13

Let V, w be vector spaces
A function L:V->w is called a linear mapping if the following holds:
Given any x,yEV and b,cER
L(bx+cy) = bL(x)+cL(y)

Question 14

Def: standard matrix

Answer 14

[L] = [Le1 … Len]

Question 15

Theorem 3.2 smth about functions

Answer 15

Given AEMmxn(R), the function f: Rn->Rm given by f(x) = Ax is a linear map

Question 16

Theorem 3.2.2

Answer 16

Let L: R^n->R^m be a linear mapping
Then L(x) = A(x) for all xER^n, where A=[L]=[Le1 ... L(en)]
Ie columns of A are L(ej)s

Question 17

Theorem 3.2.3

Answer 17

Let Rθ: R^2->R^2 be the rotation of a vector in R^2 about 0 by the angle θ
Rθ preserves addition and scalar multiplication and hence is a linear map
This is bc Rθ doesn’t change length of vectors and angles between vectors

Question 18

Def: reflection

Answer 18

Let P be a hyperplane (ie (n-1)-plane) in R^n through 0 and with normal vector n
The reflection about P is the function reflp(x) = x-2projn(x) that sends x to its mirror image across P

Question 19

Theorem 3.2 (refl Lin map)

Answer 19

reflp: R^n -> R^n, given by reflp(x), is a linear mapping

Question 20

Def: determinant

Answer 20

Given [ab] EM2x2 (R), it’s determinant is
[cd]
det[ab] = |ab| = ad-bc ER
[cd] |cd|

Only defined for square matrices