Consider the Euclidean vector space R3 with the dot product. A subspace U C R³ and vector x E R³ given by are U = span 2 ,x = 3 3 (a) Show that x ¢ U. (b) Determine the orthogonal projection TU(x) of x onto U. Show that TU(x) can be written as a linear combination of [1, 1, 1]T and [2, 2, 3]T. (c) Determine the distance d(x,U).
Consider the Euclidean vector space R3 with the dot product. A subspace U C R³ and vector x E R³ given by are U = span 2 ,x = 3 3 (a) Show that x ¢ U. (b) Determine the orthogonal projection TU(x) of x onto U. Show that TU(x) can be written as a linear combination of [1, 1, 1]T and [2, 2, 3]T. (c) Determine the distance d(x,U).
Consider the Euclidean vector space R3 with the dot product. A subspace U C R³ and vector x E R³ given by are U = span 2 ,x = 3 3 (a) Show that x ¢ U. (b) Determine the orthogonal projection TU(x) of x onto U. Show that TU(x) can be written as a linear combination of [1, 1, 1]T and [2, 2, 3]T. (c) Determine the distance d(x,U).
Consider the Euclidean vector space R3 with the dot product. A subspace U CR3 and vector x € R3 are given by U = span {0 0 ,X= 0 (a) Show that x&U. (b) Determine the orthogonal projection au(x) of x onto U. Show that au(x) can be written as a linear combination of (1,1,1]T and (2,2, 3]. (c) Determine the distance d(x,U).
Transcribed Image Text:Consider the Euclidean vector space R³ with the dot product. A subspace U C R³ and vector x E R³ are
given by
| )~日
U = span
,X =
3
(a) Show that x 4 U.
(b) Determine the orthogonal projection TU (x) of x onto U. Show that TU(X) can be written as a linear
combination of [1, 1, 1]T and [2, 2, 3]".
(c) Determine the distance d(x,U).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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![Consider the Euclidean vector space R³ with the dot product. A subspace U C R³ and vector x E R³ are
given by
| )~日
U = span
,X =
3
(a) Show that x 4 U.
(b) Determine the orthogonal projection TU (x) of x onto U. Show that TU(X) can be written as a linear
combination of [1, 1, 1]T and [2, 2, 3]".
(c) Determine the distance d(x,U).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50a58d06-8a09-44dc-be2a-000d4e5481a9%2F054efb80-2284-43d9-8d34-5618a2735956%2Frq3jpgq1m_processed.png&w=3840&q=75)
