Question

1. (a) Let V ? ?³ be a proper subspace of ?³ containing the vectors (1, 1, -4), (1, -2, 2), (-3, -3, 12), (-1, 2, -2). (i) Find a basis for V. (ii) Find the number a ? ? such that the vector u = (2, 2, a) is orthogonal to V. (b) Let W = span{(1, 2, 1), (0, -1, 2)}. (i) Determine an orthonormal basis for W. (ii) Compute pr_W((1, 1, 1)).

          1. (a) Let V ? ?³ be a proper subspace of ?³ containing the vectors
(1, 1, -4), (1, -2, 2), (-3, -3, 12), (-1, 2, -2).
(i) Find a basis for V.
(ii) Find the number a ? ? such that the vector u = (2, 2, a) is orthogonal to V.
(b) Let W = span{(1, 2, 1), (0, -1, 2)}.
(i) Determine an orthonormal basis for W.
(ii) Compute pr_W((1, 1, 1)).

1. (a) Let V ? ?³ be a proper subspace of ?³ containing the vectors
(1, 1, -4), (1, -2, 2), (-3, -3, 12), (-1, 2, -2).
(i) Find a basis for V.
(ii) Find the number a ? ? such that the vector u = (2, 2, a) is orthogonal to V.
(b) Let W = span(1, 2, 1), (0, -1, 2).
(i) Determine an orthonormal basis for W.
(ii) Compute prW((1, 1, 1)).

Added by Carmen R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/19/2023

Step 1

We can see that the given vectors are linearly dependent, since the third vector is equal to the first vector multiplied by -3, and the fourth vector is equal to the second vector multiplied by -1. So, we can remove the third and fourth vectors and still span the Show more…

Show all steps

Thanks for your feedback!

1. (a) Let V ⊂ ℐ³ be a proper subspace of ℐ³ containing the vectors (1, 1, -4), (1, -2, 2), (-3, -3, 12), (-1, 2, -2). (i) Find a basis for V. (ii) Find the number a ∈ ℐ such that the vector u = (2, 2, a) is orthogonal to V. (b) Let W = span{(1, 2, 1), (0, -1, 2)}. (i) Determine an orthonormal basis for W. (ii) Compute pr_W((1, 1, 1)).