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midterm2_review

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

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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
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(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 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. False - ( 0 1 0 0 ) 2 = ( 0 0 0 0 ) (i) L A + B = L A + L B . True - ( A + B ) v = Av + Bv (j) If A is square and A ij = δ ij for all i and j , then A = I . True - definition of the Kronecker delta (page 89) Section 2.4. V, W are finite dimensional vector spaces with ordered bases α, β , T : V W is a linear transformation, and A and B are matrices (a) [ T ] β α - 1 = T - 1 β α .
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