Let V be a vector space, and T :V →V a linear transformation such that T(5v1 + 302) = 40, – 402 and T(301 + 202) = 201 + 579. Then V2, T(3,) T(v2) T(201 – 202) =

Let B= (b,, b,, b,} and D= (d,, d,} be bases for vector spaces V and W, respectively. Let T :V- W be a linear transformation with the following properties. T(b,) = - 3d, +2d2, T(b2) = - 4d, + 4d2, T(b3) = 5d2 Find the matrix for T relative to B and D. The matrix for T relative to B and D isO.

Let V be a vector space, and TV →Va linear transformation such that T(271 +372) = -201 +502 and T(371 +50₂) = 371 - 572. Then T(v₁) =₁+₂, T(₂)=₁+₂, 40₂) = ₁+ T(471

Suppose that f : R² → R² is the linear transformation which first rotates a vector 90 degrees clock- wise and then multiplies the first coordinate by 2. Find a 2 × 2 matrix A such that f(x) = Ax for any vector x E R².

Let T: R² →R² be a linear transformation such that T (×₁,×₂) = (x₁ +×₂, 6x₁ +5x2). Find x such that T(x) = (1,12). X =

Suppose T: R¹ → R™ is a linear transformation and {V₁, V2, V3}) is a set of vectors in R". (a) Complete the following definition: The set {V1, V2, V3} is called linearly independent if (b) (c) T is one-to-one if and only if Ker (T) = Show that if T is one-to-one and {V₁, V2, V3} is linearly independent, then the set {T(V₁), T(V₂), T(V3)} is linearly independent.