Let V and W be vector spaces with ordered bases E and F, respectively. If L : V → W is a linear transformation and A is the matrix representing L relative to E and F, show that (a) v ∈ ker(L) if and only if [v]E ∈ N(A). (b) w ∈ L (V) if and only if [w]F is in the column space of A.

Suppose T: R¹ → R™ is a linear transformation and {V₁, V2, V3}) is a set of vectors in R". (a) Complete the following definition: The set {V1, V2, V3} is called linearly independent if (b) (c) T is one-to-one if and only if Ker (T) = Show that if T is one-to-one and {V₁, V2, V3} is linearly independent, then the set {T(V₁), T(V₂), T(V3)} is linearly independent.

Let G be the function from the space of 2 × 2 matrices to itself defined by: a b G (2) - [d 3a+c -b Prove that G is a linear transformation.

1. Let W be the line y = x in R². (a) Find a unit vector u on W. (b) Consider the linear transformation T: R² R² that projects each vector orthogonally onto W. Calculate the matrix for T which is uu¹. (c) Use that matrix to calculate T (d) Draw the vector [] [4] T[A] the line W, and the projection T