Prove that is U1 and U2 are subspaces of V then dim(U1 + U2) dim(U1)+dim(U2)-dim(U1 N U2). Using this, deduce that if U1 and Uz are finite dimensional and V = U1 + U, then / is the direct sum of U1 and U2 if and only if dim(V)= dim(U1)+dim(U2).
Prove that is U1 and U2 are subspaces of V then dim(U1 + U2) dim(U1)+dim(U2)-dim(U1 N U2). Using this, deduce that if U1 and Uz are finite dimensional and V = U1 + U, then / is the direct sum of U1 and U2 if and only if dim(V)= dim(U1)+dim(U2).
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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![Prove that if \( U_1 \) and \( U_2 \) are subspaces of \( V \) then
\[
\dim(U_1 + U_2) = \dim(U_1) + \dim(U_2) - \dim(U_1 \cap U_2).
\]
Using this, deduce that if \( U_1 \) and \( U_2 \) are finite dimensional and \( V = U_1 + U_2 \) then \( V \) is the direct sum of \( U_1 \) and \( U_2 \) if and only if \(\dim(V) = \dim(U_1) + \dim(U_2) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6df0d7cf-37bf-48ea-ad68-5140fb9f55b5%2Ffae745d4-78c9-43f3-9f7f-570b39bba07c%2Fxp94bni_processed.png&w=3840&q=75)
Transcribed Image Text:Prove that if \( U_1 \) and \( U_2 \) are subspaces of \( V \) then
\[
\dim(U_1 + U_2) = \dim(U_1) + \dim(U_2) - \dim(U_1 \cap U_2).
\]
Using this, deduce that if \( U_1 \) and \( U_2 \) are finite dimensional and \( V = U_1 + U_2 \) then \( V \) is the direct sum of \( U_1 \) and \( U_2 \) if and only if \(\dim(V) = \dim(U_1) + \dim(U_2) \).
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