(a) Let λ be an eigenvalue of A. Explain why a set of basic X-eigenvectors is linearly independent. (Hint: Use part (b) of the previous question.) (b) Conclude from the previous part that if A has exactly one distinct eigenvalue, and n basic eigenvectors for that eigenvalue, then the n x n matrix P with those basic eigenvectors as its columns is invertible. (Hint: Use one of the properties of a matrix you know is equivalent to invertibility.) (c) Let U and V be subspaces of R" such that U^V = {0}. Show that if S₁ = { V₁, …., Uk } is a linearly independent subset of U and S₂ = {W₁,..., Wm } is a linearly independent subset of V, then S₁ U S₂ is a linearly independent subset of Rn. (Hint: Use the definition of linear independence! Suppose that some linear combination of the vectors in S₁ U S₂ equals 0. Derive from this that a linear combination of vectors from S₁ equals a linear combination of vectors from S₂. Then use the fact that UnV = {0} along with the linear independence of S₁ and S₂ to show that all the coefficients must be Os.) (d) Let λ₁ and ₂ be two distinct eigenvalues of A. Show that it's impossible for one vector & to be both a A₁-eigenvector and a 2-eigenvector. Conclude that null(\₁ In − A) ^ null(A₂In − A) = {Ổ}. (e) Suppose A has only two distinct eigenvalues X₁ and ₂. Let S₁ and S₂ be the sets of basic X₁- and A2-eigenvectors, respectively. Use the results of the previous parts to show that S₁ U S₂ is linearly independent. (f) Conclude from the previous part that if A has only two distinct eigenvalues and n basic eigenvectors total, then the n x n matrix P with those basic eigenvectors as its columns is invertible.
(a) Let λ be an eigenvalue of A. Explain why a set of basic X-eigenvectors is linearly independent. (Hint: Use part (b) of the previous question.) (b) Conclude from the previous part that if A has exactly one distinct eigenvalue, and n basic eigenvectors for that eigenvalue, then the n x n matrix P with those basic eigenvectors as its columns is invertible. (Hint: Use one of the properties of a matrix you know is equivalent to invertibility.) (c) Let U and V be subspaces of R" such that U^V = {0}. Show that if S₁ = { V₁, …., Uk } is a linearly independent subset of U and S₂ = {W₁,..., Wm } is a linearly independent subset of V, then S₁ U S₂ is a linearly independent subset of Rn. (Hint: Use the definition of linear independence! Suppose that some linear combination of the vectors in S₁ U S₂ equals 0. Derive from this that a linear combination of vectors from S₁ equals a linear combination of vectors from S₂. Then use the fact that UnV = {0} along with the linear independence of S₁ and S₂ to show that all the coefficients must be Os.) (d) Let λ₁ and ₂ be two distinct eigenvalues of A. Show that it's impossible for one vector & to be both a A₁-eigenvector and a 2-eigenvector. Conclude that null(\₁ In − A) ^ null(A₂In − A) = {Ổ}. (e) Suppose A has only two distinct eigenvalues X₁ and ₂. Let S₁ and S₂ be the sets of basic X₁- and A2-eigenvectors, respectively. Use the results of the previous parts to show that S₁ U S₂ is linearly independent. (f) Conclude from the previous part that if A has only two distinct eigenvalues and n basic eigenvectors total, then the n x n matrix P with those basic eigenvectors as its columns is invertible.
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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Transcribed Image Text:(a) Let λ be an eigenvalue of A. Explain why a set of basic X-eigenvectors is linearly independent.
(Hint: Use part (b) of the previous question.)
(b) Conclude from the previous part that if A has exactly one distinct eigenvalue, and n basic eigenvectors
for that eigenvalue, then the n × n matrix P with those basic eigenvectors as its columns is invertible.
(Hint: Use one of the properties of a matrix you know is equivalent to invertibility.)
(c) Let U and V be subspaces of R" such that UnV = {0}. Show that if S₁ = { V₁, ..., Uk } is a linearly
independent subset of U and S₂ = {₁,. Wm} is a linearly independent subset of V, then S₁ U S₂ is
a linearly independent subset of Rn.
(Hint: Use the definition of linear independence! Suppose that some linear combination of the vectors
in S₁ U S₂ equals 0. Derive from this that a linear combination of vectors from S₁ equals a linear
combination of vectors from S₂. Then use the fact that UnV = {0} along with the linear independence
of S₁ and S₂ to show that all the coefficients must be Os.)
(d) Let λ₁ and ₂ be two distinct eigenvalues of A. Show that it's impossible for one vector to be both a
A₁-eigenvector and a A2-eigenvector. Conclude that null(λ₁ In − A) null(λ₂ In A) = {0}.
(e) Suppose A has only two distinct eigenvalues λ₁ and λ₂. Let S₁ and S₂ be the sets of basic λ₁- and
A2-eigenvectors, respectively. Use the results of the previous parts to show that S₁ U S₂ is linearly
independent.
(f) Conclude from the previous part that if A has only two distinct eigenvalues and n basic eigenvectors
total, then the n x n matrix P with those basic eigenvectors as its columns is invertible.
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could you do part f, doesnt have to be detailed
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