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University of Victoria Mathematics 211 M. Laca HW # 2 Summer 2017 Due in class Thursday 25 May 1. Find the distance 1, 1) to the plane through the origin determined by  from  the point  P (4,  1 −5 the vectors ~u =  3  and ~v =  1 . 1 −2 2. For each of the following systems of linear equations: (a) Write the augmented matrix. (b) Obtain a row equivalent matrix in row echelon form. (c) Determine whether the system is consistent or inconsistent. (d) If the system is consistent, write the general solution in standard form.     6 = x +2y +2z −5w −x3 = 6   2x1 +x2 −1 = −x −2y −z +w x1 −2x2 −2x3 = 1 (I) (II) 9 = 4x +8y +5z −8w    −x1 +12x2 +8x3 = 6  7 = 3x +6y +z −5w   1 1 2 1 3. Determine the RREF and the rank of the matrix  2 1 4 3 . 0 3 2 1        4. Show that the set         1 0 1 1 0       ,     1 2 0 0 1       ,       0 1 1 2 −2     ,     1 −3 1 0 1         is linearly dependent and find all     possible ways of writing ~0 as a linear combination of the given vectors (your answer should describe all possible choices of coefficients (α, β, γ, δ) such that ...).     3 3 −5 −6 5. Express ~b =  0  as a linear combination of the columns of A =  0 −1 −1 , or −2 −4 −3 4 show that it is impossible to do so.  6. Let A = 1 0 −1 2 3 4   , B= 4 −1 0 3 −2 3   , 2 C= 5 −1  −3 0 1 −1 −4 2  , 0 0 2   2 D =  −1  . 3 Perform each of the following matrix operations, or indicate that the operation is not defined: a) A + B (b) A + C (c) AB (d) AC (e) DT D (f) DDT  2  0 7. Let fA be the matrix mapping corresponding to A =   5 2 1 2 7 4  0 3  . 9  8 (a) What are the domain and the codomain of fA ? (that is, fA maps what into what?) .  (b) Determine  fA (  −4 2 1   )  3   and fA ( −3 ).  2         (c) Find the images of the standard basis vectors under fA . (d) Write down fA (~x) with formulas. (Your answer should give formulas for the components of fA (~x) in terms of the components of ~x). ... This question has not been answered.

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