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This question concerns the following two subsets of R4: S = {(0, 1, 3, 2), (1, 0, 1, 0), (1,-1,-1, −1)}, T= {(a, b, a+b+c, c): a, b, c = R}. (a) Show that SCT, and write down a vector in R4 that does not belong to T. (b) Show that T is a subspace of R4. (c) Show that S is a basis for T, and state the dimension of T.

This question concerns the following two subsets of R4: S = {(0, 1, 3, 2), (1, 0, 1, 0), (1,-1,-1, −1)}, T= {(a, b, a+b+c, c): a, b, c = R}. (a) Show that SCT, and write down a vector in R4 that does not belong to T. (b) Show that T is a subspace of R4. (c) Show that S is a basis for T, and state the dimension of T.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Parts a,b,c please.
This question concerns the following two subsets of R4: S = {(0, 1, 3, 2), (1, 0, 1, 0), (1,-1,-1, −1)}, T= {(a, b, a+b+c, c) : a, b, c = R}. (a) Show that SCT, and write down a vector in R4 that does not belong to T. (b) Show that T is a subspace of R4. (c) Show that S is a basis for T, and state the dimension of T. (d) Show that two of the vectors in S are orthogonal. Without using Gram-Schmidt orthogonalisation, find an orthogonal basis for T. (e) Find an expression for the vector (3, 1,-1,-5) in T as a linear combination of the vectors in your orthogonal basis for T.
Transcribed Image Text:This question concerns the following two subsets of R4: S = {(0, 1, 3, 2), (1, 0, 1, 0), (1,-1,-1, −1)}, T= {(a, b, a+b+c, c) : a, b, c = R}. (a) Show that SCT, and write down a vector in R4 that does not belong to T. (b) Show that T is a subspace of R4. (c) Show that S is a basis for T, and state the dimension of T. (d) Show that two of the vectors in S are orthogonal. Without using Gram-Schmidt orthogonalisation, find an orthogonal basis for T. (e) Find an expression for the vector (3, 1,-1,-5) in T as a linear combination of the vectors in your orthogonal basis for T.
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