By Spencer, Donald Clayton; Nickerson, Helen Kelsall

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**Example text**

If T is in- as an image, so T be linear and such that 39 ker T = 'av· T(A) = T(A' ). A = A'. 2. v are vectors of V such that Suppose A, A' A - A' e ker T = 1 t\,; hence T is injective. Proposition. Let T V -> W be linear, where are finite dimensional. (i) If (ii) If T is surjective, then (iii) If w. dim v ':?. dim w. dim v = dim w. T is injective, then dim V T is bijective, then Proof. 3. and T : V = Theorem. and 1 dim ( im T ) im T = W, ~ dim W. and dim (ker T) + dim If V and W~ dim W. e. bijective) i f and only if Proof.

Suppose A = Ei=txiAl and A = Ei=lyiAi are two representations. Then E~= 1 (xi - yi)Ai = ~. Since A1, ••• , Ak are independent, each coefficient must be zero. Therefore xi = Yi for each i, and the representation is unique. §11. 1. If D' ( D ( V, Exercises and i f D' is dependent, show that D is also dependent. 2. If a £ R, 'b £ R, (o, 1 ), (a, b) 3. in R2 Let show that the three vectors ( 1, o ), form a dependent set. ~ show by example that a basis for V need not contain u. a basis for 4. Rk, Find a basis for thus proving k = dim Rk .

Show that ~· PQ = QP Let 4. dim V be finite. Show that any endomorphism T of V can be expressed as a composition SP where projection and 1f JJ S is an automorphism. An endomorphism 5. = I. Show that, 1f is an involution. Q P is a J : V -> V is called an involution is a projection, then J = I - 2P P What involution corresponds to the projection = I - P? Ishow 6. S + T and that the set S and If A(V, W) k aT, for a e R, are affine. V -> W, In this way of all affine transformations is a vector space containing L (V, W) dimensions T are affine transformations and as a linear subspace.