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**Unformatted text preview: **MAT 442: Test 1 Solutions Throughout, F is a field. 1. Suppose V is a vector space over F . Prove that for all v ∈ V , F v = where F denotes the zero element of the field F , and denotes the zero element of V . (This is a theorem from the course.) For all v ∈ V , F v = (0 F + 0 F ) v = 0 F v + 0 F v . Now add- F v to both sides to get =- F v + 0 F v =- F v + (0 F v + 0 F v ) = (- F v + 0 F v ) + 0 F v = + 0 F v = 0 F v 2. (a) State the Dimension Theorem (also known as the Rank-Nullity Theorem). See the text. (b) Prove this theorem. (This is a theorem from the course.) See the text, Thm. 2.3. 3. Suppose T : V → W is a linear transformation, and S ⊆ V . Prove T (span( S )) = span( T ( S )) (This was proved in the course. Prove it from the definition of span and linear trans- formation.) If S = ∅ , then T (span( ∅ )) = T ( { } ) = { } and span( T ( ∅ ) = span( ∅ ) = { } . So, T (span( S )) = span( T ( S )). Now we consider the main case where S 6 = ∅ ....

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