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I) Consider V is an inner product space, and U C V a subspace. If the orthogonal projection Prv of ve V is 0, then v E U+. II) Consider R and R', with their Euclidean inner products. If T : R* → R' is is such that dim(mull T) = 2, then the orthogonal complement (range T)+ of the range of T has dimension 2. III) If the n-by-n matrix A is diagonalizable with all eigenvalues equal to 3, then A = 31, where I is the identity matrix.

I) Consider V is an inner product space, and U C V a subspace. If the orthogonal projection Prv of ve V is 0, then v E U+. II) Consider R and R', with their Euclidean inner products. If T : R* → R' is is such that dim(mull T) = 2, then the orthogonal complement (range T)+ of the range of T has dimension 2. III) If the n-by-n matrix A is diagonalizable with all eigenvalues equal to 3, then A = 31, where I is the identity matrix.

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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I) Consider V is an inner product space, and U C V a subspace. If the orthogonal projection Prv of v € V is 0, then v E U+. II) Consider R and Rª, with their Euclidean inner products. If T : R → R' is is such that dim(nullT) = 2, then the orthogonal complement (range T)- of the range of T has dimension 2. III) If the n-by-n matrix A is diagonalizable with all eigenvalues equal to 3, then A = 31, where I is the identity matrix. IV) A matrix in Mat33(R) can have eigenvalues 3, 5, and 2+ i. V) Every matrix A, € Mat3,3(R) with characteristic polynomial r* – 2r2 – r is diago- nalizable.. 1
Transcribed Image Text:I) Consider V is an inner product space, and U C V a subspace. If the orthogonal projection Prv of v € V is 0, then v E U+. II) Consider R and Rª, with their Euclidean inner products. If T : R → R' is is such that dim(nullT) = 2, then the orthogonal complement (range T)- of the range of T has dimension 2. III) If the n-by-n matrix A is diagonalizable with all eigenvalues equal to 3, then A = 31, where I is the identity matrix. IV) A matrix in Mat33(R) can have eigenvalues 3, 5, and 2+ i. V) Every matrix A, € Mat3,3(R) with characteristic polynomial r* – 2r2 – r is diago- nalizable.. 1
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