midterm2_review

midterm2_review - Midterm 2 Review Section 2.2. V,W are...

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Unformatted text preview: Midterm 2 Review Section 2.2. V,W are finite dimensional vector spaces with ordered bases β,γ , and T,U : V → W are linear transformations (a) For any scalar a , aT + U is a linear transformation from V to W . • True - Theorem 2 . 7(a), or just check that ( aT + U )( cv + w ) = c ( aT + U )( v ) + ( aT + U )( w ) (b) [ T ] γ β = [ U ] γ β implies that T = U . • True - linear transformations are completely determined on a given basis (c) If m = dim( V ) and n = dim( W ), then [ T ] γ β is an m × n matrix. • False - [ T ] γ β is an n × m matrix (remember that [ v ] β is an m × 1 matrix and [ T ( v )] γ is an n × 1 matrix) (d) [ T + U ] γ β = [ T ] γ β + [ U ] γ β . • True - Theorem 2 . 8(a) (e) L ( V,W ) is a vector space. • True - see the two definitions on page 82 (f) L ( V,W ) = L ( W,V ). • False - if V and W are different sets, then a function from V to W cannot be a function from W to V- see the definition of functions Section 2.3. V,W,Z are finite dimensional vector spaces with ordered bases α,β,γ , and T : V → W and U : W → Z are linear transformations, and A and B are matri- ces (a) [ UT ] γ α = [ T ] β α [ U ] γ β . • False - [ UT ] γ α = [ U ] γ β [ T ] β α (see Theorem 2 . 11) (b) [ T ( v )] β = [ T ] β α [ v ] α for all v ∈ V . • True - Theorem 2 . 14 (c) [ U ( w )] β = [ U ] β α [ w ] β for all w ∈ W . • False - [ U ] β α doesn’t even make sense since α is a basis for V , not W (d) [ I V ] α = I . • True - definitions (e) T 2 β α = [ T ] β α 2 . • False - T 2 doesn’t make sense unless W = V (f) A 2 = I implies that A = I or A =- I . • False - ( 1 0- 1 ) 2 = I (g) T = L A for some matrix A . • False - T is defined on V , L A is defined on F n (h) A 2 = 0 implies that A = 0, where 0 denotes the zero matrix....
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This note was uploaded on 03/06/2010 for the course ECON 102 taught by Professor Serra during the Spring '08 term at UCLA.

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midterm2_review - Midterm 2 Review Section 2.2. V,W are...

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