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Assignment 3 - (b Show that the y-z plane is not ^invariant...

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v f *• 1. Find the matrix representations of the linear transformations below with respect to the given bases. = \1x - 3y x + y - z o/ \i (n),[7T, 7 = 0 , 1 , 1 \0/ VO/ ll (i) AT]", n = T : P Z (R) - P 2 (R), T(a + bx + ex 2 ) = (a + 26 + 2c) + (2a + ft + 2c)x + (2o + 26 + c)x 2. Use your solutions to Question 1 to evaluate (i (b) (i) and T( I + 2r + 3i a ) and T(2 + 2l - 2l a ).
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J Consider the linear transformations T : R 2 -> P 2 (M) and S : ft(R) • M 2X2 (R) given by T((a, b) T ) = (a 4- 26) + (-a + 3b)x 4- (3a - 26)i 2 , <..,„>-/'PW+P'W *) {83690983}- Find (a) T [T]", (b) ,[Sp, and find (c) j[SoT]" two different ways. ^._ Let S : R 3 -i K :| be given by M /f, ..i ^ \ /2a +J6 + 3c\ = 3d + 2)M \3a+36 + 2c and let /? and 7 be the ordered bases (a) (b) (c) f 0= \ \ \ 1 ( l \ ^\\} Find alS] 1 *. Find ,,[/p and ,[/]«. /1\ I n . \o/ 1 / '7i( M i W ' i -i ^ o Evaluate 7 [5] 7 two different ways. 0' 0 1 ±( ; V « l _ J
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j~ Let L : V * V be a linear operator. Show that the following subspaces of V are L-invariant: (a) V, (b) {<V}, (c) M(L), (d) K(L). £ Let L : R 3 -> K 3 be given by i(l, y, 3) = (i + y + z, 2 - y, z + 2j/). (a) Show that the z-axis is L-invariant. (b} Show that the y-z plane is not ^invariant. 7 Let T : V V be a linear operator and let v V satisfy T(v) mv where
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Unformatted text preview: (b} Show that the y-z plane is not ^invariant. 7 Let T : V -» V be a linear operator and let v € V satisfy T(v) — mv where m is a fixed element of the field F. Now define the subset En = (u e V : T(u) = mil}. (a) Show that E m is a vector subspace of V. (b) Show that E m is T-invariant. g Let L : V — * V be a linear Operator on the n-dimcnsioim] vector space V, and let v £ V. (a) Let m. be the smallest integer with the property that L m (v) E Span({v,i(v) L m-'(v)}. Prove that S. = Span({v, i(v),. .., L m-'(v)}). C**""<ir Kl»u.) (b) Let L : M 2x2 fft) --> A/ 2x2 (K) be given by Find a basis for the /^invariant subspace generated by 1 M oo ( l 001 1 1 1 Definition. Let L : V — * V be a linear operator on a vector space V and let v £ V. The L-cyclic subspace of V generated by v, S v , is defined by 5, = Span({v, i(v), L 2 (v) ^*( v ), •• •})...
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