All you need to show is that $T$ satisfies $T(cA+B) =cT(A) +T(B)$ for any vectors $A,B$ in $\mathbb{R}^4$ and any scalar from the field, and $T(0) =0$. It looks like you got it. That should be sufficient proof.

Our Denver off-road shop offers top-notch installation services to ensure your vehicle is adventure-ready. Trust us to provide quality parts and exceptional service. Explore more & get your gear installed by experts today!

Jul 10, 2018 · Give a basis for $\mathrm{Ker}(T)$ and $\mathrm{Im}(T)$ of the linear transformations

Determine whether the linear transformation is one-to-one, onto, or neither. (Select all that apply.) T: R4 → R4, T (X, Y, Z, W) = (V, W, X, z) O one-to-one onto neither. Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on.

Mar 29, 2025 · Let T be a linear transformation from V to W i.e T: V → W and V is a finite-dimensional vector space then Rank (T) + Nullity (T) = dim V. Analysis: Given: T : R 4 → R 4. T(x, y, z, u) = (x, y, 0, 0) Let us first find the dimension of null space (nullty) Null space of T = kernel of T (ker T) Ker T = { v ∈ V | T(v) = 0} Ker T = (0, 0, 0, 0 ...

By choosing correctly $M \in GL(4,\mathbb C)$, you can make the operator $T=M^{-1}AM$ real: $T : \mathbb R^4 \longrightarrow \mathbb R^4$.

Then the columns of [T] are the vectors obtained when T acts on each of the standard basis vectors e1, e2, ... , e m. We indicate this by [T] = [Te1 Te2 ··· Te m] ⋄ Example 10.2(g): Let T be the transformation in R2 that rotates all vectors counterclockwise by ninety degrees.

We have given that the linear transformation T: R 4 → R 4 defined as T (x, y, z, w) = (y, z, x, w) To find kerT . Let, k e r T = {X ∈ R 4: T (X) = 0}

Consider the linear transformation T: R4 → R4 defined by T(w, x, y, z) = (0, 2w, 3w + 2x, x + 4y). a) Show that T is a linear transformation and verify the rank-nullity theorem b) Find the nilpotence of T.

Nov 3, 2019 · Construct a linear transformation T : R4 → R4 such that Kernel(T) = Image(T). How about the same for a linear transformation S : R5 →R5.