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3. Let V, W and U be finite dimensional vector spaces. Suppose S: V W andT:WU are linear. Show (a) null(T S) Snull(T) + null(S) (b) rank(T S) < min(rank(T ), rank(S)) Hint: Consider the linear transformation T|im(S): im(S)U, the restriction of T to im(S).

3. Let V, W and U be finite dimensional vector spaces. Suppose S: V W andT:WU are linear. Show (a) null(T S) Snull(T) + null(S) (b) rank(T S) < min(rank(T ), rank(S)) Hint: Consider the linear transformation T|im(S): im(S)U, the restriction of T to im(S).

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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3. Let V, W and U be finite dimensional vector spaces. Suppose S: V W andT:W U are linear. Show (a) null(T S)< null(T ) + null(S) (b) rank(T S) <min(rank(T ), rank(S) Hint: Consider the linear transformation Tim(S): im(S) U, the restriction of T to im(S).
Transcribed Image Text:3. Let V, W and U be finite dimensional vector spaces. Suppose S: V W andT:W U are linear. Show (a) null(T S)< null(T ) + null(S) (b) rank(T S) <min(rank(T ), rank(S) Hint: Consider the linear transformation Tim(S): im(S) U, the restriction of T to im(S).
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