Definition: A matrix transformation T : R n → Rm is said to be onto if
... Definition: A matrix transformation T : Rn → Rm is said to be onto if evey vector in Rm is the image of at least one vector in Rn . Theorem 8.2.2: If T is a matrix transformation, T : Rn −→ Rn , then the following are equivalent (a) T is one-to-one (b) T is onto Example: Is the matrix transformation ...
... Definition: A matrix transformation T : Rn → Rm is said to be onto if evey vector in Rm is the image of at least one vector in Rn . Theorem 8.2.2: If T is a matrix transformation, T : Rn −→ Rn , then the following are equivalent (a) T is one-to-one (b) T is onto Example: Is the matrix transformation ...
HOMEWORK 1 SOLUTIONS Levandosky, Linear Algebra 1.2 (a
... a(u + v) + b(u + w) + c(v + w) = 0. We must show that a, b, and c are all equal to 0 (since this is the very definition of linear independence!). Note that we can rewrite the above equation as (a + b)u + (a + c)v + (b + c)w = 0. Since {u, v, w} is a linearly independent set by hypothesis, the above ...
... a(u + v) + b(u + w) + c(v + w) = 0. We must show that a, b, and c are all equal to 0 (since this is the very definition of linear independence!). Note that we can rewrite the above equation as (a + b)u + (a + c)v + (b + c)w = 0. Since {u, v, w} is a linearly independent set by hypothesis, the above ...
8. Graphing Simple Rational Functions
... touch the dotted line. The dotted x = -4 line the branches never touch is called an ‘asymptote.’ ...
... touch the dotted line. The dotted x = -4 line the branches never touch is called an ‘asymptote.’ ...
