V is a 4-dimensional vector space with basis B = (v1, V2 , V3 , V4). T:V → V is linear transformation with rank three and satisfies T(v1) = v + v2 +- v3 + v4, T(v2) = v1 + 2v1 + 3v3 + 4v4, T(v3) = v1 + 3v2 + 4v3 + 6v4. If T(v4) = v1 + 4v2 + 5v3 + avĄ what is the value of a? 8 O 7 O 5

Show that the vectors X1 = 2e2t and 2et X2 5e3t form a linearly independent set.

Find an orthonormal basis of the plane x1 + 5x2 – x3 = 0.

Consider the vectors u₁ = [1,0,−1], _U2 = [1,0,0] and u3 = [0, 1, 1], v = [5, 1, 3] and w = = [1,0,3]. (a) Show that the vectors u₁, u₂ and u3 are linearly independent. (b) Show that the vector v is in the span of u₁, U₂ and u3. (c) Find the coordinate vector of v with respect to {U₁, U2, U3}. (d) Write the zero vector as as a non-trivial linear combination of the vectors u₁, U2, U3 and w.