Show each of the following maps is linear, describe the kernel and image, and
compute dim ker T and dim Im T when they are finite. Find a basis for finite-
dimensional kernels and images.
a Given P is an invertible n n matrix, T: M_n n(R) → M_n n(R) is defined by
T(A) = P−!AP, for A ∈ M_n n(R).
b T: R3 → R is defined by T(X) = [-2, 3, 1] X, for X ∈ R3.
c. Orthogonal Projection. T: R⁴ → R" is given by T((x, y, z, w)) = (x, y), for (x, y, z, w) ∈ R⁴.
d. Orthogonal Projection. Let V be an n-dimensional real inner product space and let W
be an m-dimensional subspace of V. Let {w₁, w₂, ..., w_m} be an orthonormal
basis for W. Define T: V → W by T(v) = ∑(v, w_i)w_i, for v ∈ V.
e T: P₂ → P₂ is defined by Tp = q, where q(x) = p(x) + p(−x), for p ∈ P₂.
f T: P₂ → R is defined by Tp = ∫₀! p(x) dx, for p ∈ P₂.
g. Let V = P₁([-1, 1]), y ∈ V given by y(t) = 1 + t, and let T: V → R be defined by
T(x) = ∫₁ⁱ x(t)y(t) dt, for x ∈ V.
h T: R3 → R3 is defined by T((x, y, z)) = (2x + z, 2y + 3z, x − y − z), for (x, y, z) ∈ R3.
Important Note: Later, we'll see we can equivalently define T: R3 → R3 by
T(X) = AX, for X ∈ R3, where A = [[2, 0, 1], [0, 2, 3], [1, -1, -1]].
i Let A = [[4, 1, 5, 2], [1, 2, 3, 0]] and let T: R⁴ → R" be defined by T(X) = AX, for X ∈ R⁴.
