Question

Consider the following W subspace and given vector x in the R<sup>4</sup> vectorspace: W := Span((1, 1, -1, 1), (3, 1, 2, 2), (3, -3, 3, -1)), x = (4, 7, 7, -8). (a) (2 points) Decide if the generator system of W is an orthogonal system or not. (b) (8 points) Give an orthogonal and an orthonormed basis in the W subspace. (c) (4 points) Decompose vector x into a parallel and an orthogonal component regarding subspace W.

          Consider the following W subspace and given vector x in the R<sup>4</sup> vectorspace:
W := Span((1, 1, -1, 1), (3, 1, 2, 2), (3, -3, 3, -1)),  x = (4, 7, 7, -8).
(a) (2 points) Decide if the generator system of W is an orthogonal system or not.
(b) (8 points) Give an orthogonal and an orthonormed basis in the W subspace.
(c) (4 points) Decompose vector x into a parallel and an orthogonal component regarding
subspace W.

Consider the following W subspace and given vector x in the R<sup>4</sup> vectorspace:
W := Span((1, 1, -1, 1), (3, 1, 2, 2), (3, -3, 3, -1)), x = (4, 7, 7, -8).
(a) (2 points) Decide if the generator system of W is an orthogonal system or not.
(b) (8 points) Give an orthogonal and an orthonormed basis in the W subspace.
(c) (4 points) Decompose vector x into a parallel and an orthogonal component regarding
subspace W.

Added by Joel M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

05/19/2023

Step 1

To determine if the generator system of W is orthogonal, we need to check if the dot product of any two distinct vectors in the generator system is zero. If all dot products are zero, then the generator system is orthogonal. Show more…

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Consider the following W subspace and given vector x in the R vector space: W = Span{1, 1, -1, 1, 3, 1, 2, 2, 3, -3, 3, -1} x = 4, 7, 7, -8 (a) (2 points) Decide if the generator system of W is an orthogonal system or not. (b) (8 points) Give an orthogonal and an orthonormal basis in the W subspace. (c) (4 points) Decompose vector x into a parallel and an orthogonal component regarding subspace W.