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FALSE 1 For any function defined on vector spaces, p: (V, +,) → (V', +';") Dim(V) = Nullity ofo + Rank of o TRUE %3D If 9: (V, +,) → (v', +',') is a vector homomorphism, the Null Space ofo is a subspace of (V',+','). TRUE FALSE [2] If 9: (V,+;) → (V', +',') is a vector homomorphism, the Range Space ofo is a subspace of (V',+','). TRUE FALSE [3]

FALSE 1 For any function defined on vector spaces, p: (V, +,) → (V', +';") Dim(V) = Nullity ofo + Rank of o TRUE %3D If 9: (V, +,) → (v', +',') is a vector homomorphism, the Null Space ofo is a subspace of (V',+','). TRUE FALSE [2] If 9: (V,+;) → (V', +',') is a vector homomorphism, the Range Space ofo is a subspace of (V',+','). TRUE FALSE [3]

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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**V. True/False** Circle either "true" or "false" to indicate the veracity of each statement. You do NOT have to give any reasons for your answers. 1. TRUE FALSE For any function defined on vector spaces, \( \varphi: (V, +, \cdot) \rightarrow (V', +', \cdot') \): \(\text{Dim}(V) = \text{Nullity of } \varphi + \text{ Rank of } \varphi\) 2. TRUE FALSE If \( \varphi: (V, +, \cdot) \rightarrow (V', +', \cdot') \) is a vector homomorphism, the Null Space of \( \varphi \) is a subspace of \( (V', +', \cdot') \). 3. TRUE FALSE If \( \varphi: (V, +, \cdot) \rightarrow (V', +', \cdot') \) is a vector homomorphism, the Range Space of \( \varphi \) is a subspace of \( (V', +', \cdot') \). 4. TRUE FALSE If \( \alpha: (V, +, \cdot) \rightarrow (V', +', \cdot') \) and \( \beta: (V, +, \cdot) \rightarrow (V', +', \cdot') \) are two vector homomorphisms, then \(\mathcal{R}(\alpha) = \mathcal{R}(\beta)\). 5. TRUE FALSE If \( \alpha: (V, +, \cdot) \rightarrow (V', +', \cdot') \) and \( \beta: (V, +, \cdot) \rightarrow (V', +', \cdot') \) are two vector isomorphisms, then \(\mathcal{R}(\alpha) = \mathcal{R}(\beta)\). 6. TRUE FALSE Suppose \((V, +)\) has a basis \(B = \langle \beta_1, \beta_2 \rangle\), and we write \(B' = \langle \beta_2, \beta_1 \rangle\). Let \(\vec
Transcribed Image Text:**V. True/False** Circle either "true" or "false" to indicate the veracity of each statement. You do NOT have to give any reasons for your answers. 1. TRUE FALSE For any function defined on vector spaces, \( \varphi: (V, +, \cdot) \rightarrow (V', +', \cdot') \): \(\text{Dim}(V) = \text{Nullity of } \varphi + \text{ Rank of } \varphi\) 2. TRUE FALSE If \( \varphi: (V, +, \cdot) \rightarrow (V', +', \cdot') \) is a vector homomorphism, the Null Space of \( \varphi \) is a subspace of \( (V', +', \cdot') \). 3. TRUE FALSE If \( \varphi: (V, +, \cdot) \rightarrow (V', +', \cdot') \) is a vector homomorphism, the Range Space of \( \varphi \) is a subspace of \( (V', +', \cdot') \). 4. TRUE FALSE If \( \alpha: (V, +, \cdot) \rightarrow (V', +', \cdot') \) and \( \beta: (V, +, \cdot) \rightarrow (V', +', \cdot') \) are two vector homomorphisms, then \(\mathcal{R}(\alpha) = \mathcal{R}(\beta)\). 5. TRUE FALSE If \( \alpha: (V, +, \cdot) \rightarrow (V', +', \cdot') \) and \( \beta: (V, +, \cdot) \rightarrow (V', +', \cdot') \) are two vector isomorphisms, then \(\mathcal{R}(\alpha) = \mathcal{R}(\beta)\). 6. TRUE FALSE Suppose \((V, +)\) has a basis \(B = \langle \beta_1, \beta_2 \rangle\), and we write \(B' = \langle \beta_2, \beta_1 \rangle\). Let \(\vec
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