Linear Algebra Answers

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}.

find a linear transformation T:R^3->R^3 whose image is spanned by (1,2,3 ) and (4,5,6)

. let t:r^2->r^2 be a linear transformation for which ( 1,2)= (2,3 ) and ( 0,1)= (1,4 ). find a formula for t.