Let {v1, . . . , vn} be a basis for a vector space V, and let L1 and L2 be two linear transformations mapping V into a vector space W. Show that if L1(vi) = L2(vi) for each i = 1, . . . , n, then L1 = L2 [i.e., show that L1(v) = L2(v) for all v ∈ V].

Let V be a vector space, v, u € V, and let T₁: V → V and T₂: V → V be linear transformations such that T₁(v) 6v +3u, T₁(u) = -5v – 2u, T₂(v) = 2v+3u, T₂(u) = 5v + 5u. Find the images of u and u under the composite of T₁ and T₂. (T₂T₁)(v) = (T₂T₁)(u) =

Let T: V-->W be a linear transforma­tion between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let       A= [ T]C<--B. Use the results of this section to give a matrix­based proof of the Rank Theorem

Let M₂x2 be the vector space of all 2 x 2 matrices and define a linear transformation T: M2x2 → M2x2 by a [id] Describe the kernel of T. C T(A) = A + AT, where A = O No answer text provided. odlo 0 -a ° 0 O [1] :d is a real number} [9] : b is a real number} -b O (1 :d is a real number} (d c is a real number} °41 :d is a real number} : a,b,c,d are real numbers} [9] : b is a real number}

Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1. Show that 1 (v, w1) + 2 (v, w2) + 3 (v, w3) + ...+ k (v, wk) cannot be a negative number.