find bases for the kernel and range of the linear transformations T in the indicated exercises. In each case, state the nullity and rank of T and verify the Rank Theorem Exercise 4

Finite Mathematics Chapter 3 Section 1.1 Operations Identities 0 - Zero Addition/ Subtraction 1 - One Multiplication/ Division Zero is the additive Identity as displayed above. You can add or subtract zero from any number without changing that number's value. One is the multiplicative identity as displayed above. You can multiply or divide any number by one without changing that number's value. Functions can also: o Be added or subtracted: i.e. f(x) + g(x) o Multiplied or divided: i.e: f(x) * g(x) o Make composite functions: i.e fog(x) = f(g(x)) Inverses: An inverse gets you back to the identity o Example: The additive inverse of 10 is –10 o Example: The multiplicative inverse of 4 is 4^-1 More Info: If it is a function, it will pass the vertical line test If it is an inverse, it will pass the horizontal line test There are three ways to define a line: 1. Slope-Intercept form ---> y= mx + b 2. Point-Slope Form --> y-y1 = m(x-x1) 3. 2 Points (x1, y1) (x2, y2) --> Slope: y2- y1 x2- x1 What is a matrix** A rectangular block of numbers The rows of a matrix can be: o Added o Multiplied by constants o Or switched Example of a Matrix: 1 1 3 1 -1 1 x + y = 3 x – y = 1 (the y's cancel when you add the equations together) 2x = 4 x = 2 (this gives you the x value in the coordinate) 1 1