Linear Algebra Unit 1

A=[k 0 , 0 1]

Horizontal expansion (dilation) by k

A= [1 0, 0 k]

Vertical expansion (dilation) by k

A=[1 0, 0 -1]

Reflection over the x₁ axis

A= [-1 0, 0 1]

Reflection over the x₂ axis

A=[0 1, 1 0]

Reflection over the x₁=x₂ axis

A=[0 -1, -1 0]

Reflection over x₁=-x₂ axis

A=[1 k, 0 1]

Horizontal shear of factor k

A=[1 0, k 1]

Vertical shear of factor k

A=[cos(x) -sin(x), sin(x) cos(x)]

Rotation of angle x about the origin

A=[1 0, 0 0]

Projection onto the x₁ axis

A=[0 0, 0 1]

Projection onto the x₂ axis

Homogeneous linear system

A linear system where Ax= the zero vector and A is a mxn matrix

Non homogeneous linear system

A linear system where Ax=b

Independent

A linear system where A=[a₁, a₂….] and Ax=0 vector for (xa₁….=0 vector)

Dependent

A linear system where:

-One of the vectors in the set is the linear combination of another

-There are more vectors than entries in a vector

-The set contains the 0 vector

Consistent

A system with a possible solution

Inconsistent

A system with no solution, ie [0 0|1]

One to one

A translation where:

-each vector in R^m is the image of at most 1 vector in Rⁿ

-T is linear and has only trivial solution

-A matrix is linearly independent