Let V and W be vector spaces and let T V W be a linear transforma- tion. Suppose that {T(v₁), T(v₂),..., T(vn)} is linearly independent. Prove that {V1, V2,..., Vn} is linearly independent.

7. Determine if the set S = {-1 + x², 5 + 2x} of vectors in P2 are orthogonal with (f,g) = f(x)g(x) dx.

Let V and W be finite-dimensional vector spaces and T: V →W be linear. (a) Prove that if dim(V) <dim(W), then T cannot be onto. (b) Prove that if dim(V) >dim(W), then T cannot be one-to-one.

Show that the vectors p1(x) = a + bx and p2(x) = c + dx are linearly independent in P1(R) if and only if the constants a, b, c, d satisfy ad − bc ≠ 0

Let V be the vector space of functions spanned by B = {b₁ = 2 sin x + 9 cos x, b₂ = 4 sin x + 7 cos x} where x C = {C₁ = sin x, C₂ = cos } where x of coordinates matrix P. C+B P = C←B [f(x)] c = [ The coordinates of a function f(x) relative to the basis B are [f(x)] B = [8] 6 coordinates of f(x) relative to C and find f(x). f(x) = Ex: 3 Ex: 3 nn 2 sin x + -, ne Z nn 2,n € Z is also a basis for V. Find the change COS x Find the

Let V be a finite dimensional vector space. Let T E L(V). Show that there exists an integer n such that Ker(T") = (a) Ker(T+1) = Ker(T"+2) Hint: Ker(Tk) c Ker(Tk+1) (b) .. Show that V = Ker(T") Im(T").

What is Power and Multiple Rules?

V is a 4-dimensional vector space with basis B = (v1, v2, v3, v4). Set wi = v1 + V2 + v3 + V4 , W2 = vị + 2v2 + 3v3 + 2v4. w3 = V1 + v2 + 2v3 + 4v4, w4 = v1 + 2v2 + 2v3. Then B' = (w1, w2, w3, w4) is also a basis of V. T:V → V is a linear transformation such that T(w1) = T(w2) = 0v,T(w3) = wi, T(w4) = w2. If T(v4) = avi + bv2 + cv3 + dv4 which of the following is a + b+ c+ d? O 8 10 O 5 3

Let c₁ (t) = esti + 3 sin(t)j + t7k and c₂(t) = e−³¹i + 2 cos(t)j – 8t¹k. (Enter your solution as a single vector using the vector form (*,*,*). Use symbolic notation and fractions where needed.) -[c₁ (t) + c₂ (t)] = dt

Find a vector x whose image under T, defined by T(x) = Ax, is b, and determine whether x is unique. Let 1 3 7 10 4 15 31 46 0 1 1 2 -4 -13 - 29 - 42 A = 1 X = b= Find a single vector x whose image under T is b.