TUE7Quiz5 - Introduction to Linear Algebra Quiz#5 Tue 7 pm...

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Unformatted text preview: Introduction to Linear Algebra Quiz #5 Tue. 7 pm Department I ID I Name 1 Think all vectors as column vectors. 1. Let V, W be subspaces of R”. (a) Write the definition of a basis for V. A) A set of vectors in V which is linearly independent and spans V. (b) If dim(V) S dim( W), is Va subspace of W? A) No. In R3, a line through the origin has dimension 1, and the plane through the origin and orthogonal to the line has dimension 2. But clearly, the line is not a subspace of the plane. 2. Find the canonical basis for the solution space of the homogeneous system, and state the dimension of the space. :11:1 — 372+ .733 :0 21'l * $2 "4333 =0 3301+ $2 W 11m3 =0 1*1 1 0 A) The augmented matrix of the system is [2—1 4 0 and the r.r.e.f. of the matrix is 3 1 110 1030 0120 . Thus the general solution is 1' = (—3t,—2t,t) : t(—3,—2,1) where —oo<t< 00. 0000 The canonical basis is {(i3,72,1)} and the dimension is 1. 3. Let A be a linear transformation from R" to R’”, and v1, v are linearly 7L independent vectors in 12“. What is the necessary and sufficient condition for A111, Avn are linearly independent in 13"”? A) A721, A?) are linearly independent . n (I) clAv1 + + CnAvn, = O has only trivial solution. (E) A(c1v1 + + Cn’Un) = O has only trivial solution. (E) Av = O has only trivial solution. (1) A is one to one. ...
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