Question

Let T: \(\mathbb{R}^3 \to \mathbb{R}^3\) be the linear transformation that projects u onto v = (2, -1, 1). (a) Find the rank and nullity of T. rank 1 nullity 2 (b) Find a basis for the kernel of T. \begin{bmatrix} 1/2 & 2 & 3 \\ 1/2 & 0 & 1 \end{bmatrix}

          Let T: \(\mathbb{R}^3 \to \mathbb{R}^3\) be the linear transformation that projects u onto v = (2, -1, 1).
(a) Find the rank and nullity of T.
rank 1
nullity 2
(b) Find a basis for the kernel of T.
\begin{bmatrix} 1/2 & 2 & 3 \\ 1/2 & 0 & 1 \end{bmatrix}

Let T: ℝ^3 →ℝ^3 be the linear transformation that projects u onto v = (2, -1, 1).
(a) Find the rank and nullity of T.
rank 1
nullity 2
(b) Find a basis for the kernel of T.

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Added by Kayla M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

05/19/2022

Step 1

For v = (2, -1, 1) and u in R^3, T(u) = proj_v(u) = ((u · v) / (v · v)) v. Here v · v = 2^2 + (-1)^2 + 1^2 = 6, so T(u) = ((u · v)/6) v. Show more…

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Let T: R^3 -> R^3 be the linear transformation that projects u onto v = (2, -1, 1). (a) Find the rank and nullity of T. rank 1 nullity (b) Find a basis for the kernel of T. 1/2 1/2 0 11