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**Unformatted text preview: **Math 221, Winter 2016, Section 202 Page 1 of 4 Quiz VI
February 16, 2017 No books. No notes. No calculators. No electronic devices of any kind. Name Student Number Problem 1. (8 points) In each case decide whether or not V is a subspace. If V is not a subspace, explain why not. If V is a subspace, ﬁnd a basis for V (no further explanation necessary). 1. V C R3 is the plane through the origin with equation m+3y-2z=0. \} 15 Q Maxim .6 R3. 6}an scum -. (3% (2V2t> ¢Y(. R been (4 V is (SA?) 2. V C R3 is the set of solutions to the system of linear equations 0
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3. V is the Md quadrant in R2, deﬁned to consist of all (g) E R2 such that $20andy30.
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gs \} is NOT a hbsfme J R _ 4. V is the range of the linear transformation T : R2 —> R3, given by the formula
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T (I) = x - 2y .
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It is given that 3494—3)) is a basis of R2. Note that the two vectors in B are orthogonal to each other. Denote the stande basis of R2 by S. '
Let T : R2 —> R2 be orthogonal projection onto the line L = span (—43) . 1. Find the matrix [T] ,5 of T with respect to the basis 13.
2. Find the matrix P such that [015 = P [1313, for all vectors 17 E R2.
3. Find the matrix Q such that [ﬁg = Q [1713, for all vectors 27 E R2. ’1’) 4. Find the standard matrix [T]5 of the projection T. @ (P is “a Ira-NSX'M “h”; "V, “WM“: me lea ban? WM (P :(QtS ‘1) - Lt 3 - _I_ “l '5
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