Linear Algebra Answers

Let L : R

3 −→ R3[x] be a linear transformation such that

L(1, 0, 0) = 2x + x3, L(0, 1, 0) = −2x + x2, and L(0, 0, 1) = x2 + x3

.

(a) Find a formula for L(a, b, c), where a, b, c ∈ R.

(b) Find a basis for ker L. Is L a monomorphism?

(c) Find a basis for Im L. Is L an epimorphism?

(d) Find a basis for L−1[A], where A = x^2

Find an expression of z^n which nEN

Let V and W be vector spaces over F, where V is n-dimensional. Let K ≤ V and R ≤ W

be finite-dimensional subspaces such that dim K + dim R = n. Prove that there exists a linear

transformation L : V −→ W such that ker L = K and Im L = R.

1. Let X = {v1, v2, . . . , vn} be a subset of a vector space V over F. Let

A(X) := {α1v1 + · · · + αnvn | α1 + α2 + · · · + αn = 1}.

Prove that A(X) is a subspace of V if and only if vi = 0V for some i ∈ {1, 2, . . . , n}.

Let V and W be vector spaces over F, where V is n-dimensional. Let K ≤ V and R ≤ W

be finite-dimensional subspaces such that dim K + dim R = n. Prove that there exists a linear

transformation L : V −→ W such that ker L = K and Im L = R.

Let X = {v1, v2, . . . , vn} be a subset of a vector space V over F. Let

A(X) := {α1v1 + · · · + αnvn | α1 + α2 + · · · + αn = 1}.

Prove that A(X) is a subspace of V if and only if vi = 0V for some i ∈ {1, 2, . . . , n}.

Let X = {v1, v2, . . . , vn} be a subset of a vector space V over F. Let

A(X) := {α1v1 + · · · + αnvn | α1 + α2 + · · · + αn = 1}.

Prove that A(X) is a subspace of V if and only if vi = 0V for some i ∈ {1, 2, . . . , n}.