Linear Algebra Quotes

Mathematics is the science which draws necessary conclusions.

The contribution of mathematics, and of people, is not computation but intelligence.

Matrices act. They don't just sit there.

So far our attention has focused on vector spaces. No one gets excited about vector spaces.

The reason special names are given to these quadratic irrationalities is that any quadratic algebraic integer is a linear combination (with ordinary integers as coefficients) of 1 and one of these fundamental quadratic algebraic integers.

The collection of all real or complex numbers that are integral linear combinations of 1 and τd is closed under addition, subtraction, and multiplication, and is therefore a ring, which we denote by Rd. That is, Rd is the set of all numbers of the form a + bτd where a and b are ordinary integers. These rings Rd are our first, basic, examples of rings of algebraic integers beyond that prototype, , and they are the most important rings that are receptacles for quadratic irrationalities. Every quadratic irrational algebraic integer is contained in exactly one Rd.

It is simpler to redefine the meaning of “equal.

You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast.

You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast.

Each operation contributes to AES’s security in a specific way: * Without KeyExpansion, all rounds would use the same key, K, and AES would be vulnerable to slide attacks. * Without AddRoundKey, encryption wouldn’t depend on the key; hence, anyone could decrypt any ciphertext without the key. * SubBytes brings nonlinear operations, which add cryptographic strength. Without it, AES would just be a large system of linear equations that is solvable using high-school algebra. * Without ShiftRows, changes in a given column would never affect the other columns, meaning you could break AES by building four 232 element codebooks for each column. (Remember that in a secure block cipher, flipping a bit in the input should affect all the output bits.) * Without MixColumns, changes in a byte would not affect any other bytes of the state. A chosen-plaintext attacker could then decrypt any ciphertext after storing 16 lookup tables of 256 bytes each that hold the encrypted values of each possible value of a byte.