Instructions: Attempt all problems. You should provide appropriate justification for youranswers and refrain from using formulac/results that have not been discussed in lectures.Unsubstantiated...

Answer without using advanced methods that may not be taught at undergraduate level, instructions are provided in 1.png.


Instructions: Attempt all problems. You should provide appropriate justification for your answers and refrain from using formulac/results that have not been discussed in lectures. Unsubstantiated answers will not receive full credit. 1. [10] Find, with justification, a homogencous system of lincar equations in four variables whose solution sct is equal to 2 1 5 5 So —1 0 -3 —2 SPA ge || 2 || 3 1 3 0 5 (Hint: Take a generic vector # € RY. When is # a lincar combination of the four given vectors?). 2. [12] A square matrix A is said to be idempotent if A2 = A. (a) Find all idempotent 2 x 2 matrices whose second column is the zero vector. Show your work. (b) By a result from lectures, the mapping L: R25 R? 7s proj; is linear. Find the standard matrix of L, and verify that it is idempotent. (c) Is it true that for any non-zero vector @ € R”, the standard matrix of the lincar mapping L:R" = R", Zw proj; ¥ is idempotent? Give a brief explanation of your answer. 3. [10] In cach of the following cases, determine (with justification) whether the given mapping L is lincar. In the cases where it is, find the standard matrix of L. G LR SR, | 7d" Ware, 20 zy if mma <0 (ii) l: r? — r%, [ mn ] + x11 + 250, where i and arc fixed (but unknown) ta vectors in r2. 4. [10] there is a unique lincar mapping l: r* — r? such that 1 —1 1 d1] (2d 12) wee] d120) 0 1 - —1 find, with justification, the standard matrix of l. 5. [8] the matrix 2 21 (3) 2 12 3 1 22 is the standard matrix of the lincar mapping l: r* — r? given by reflection about a plane p in 3-space that passes through the origin. find a scalar equation for p (hint: what docs l do to the vectors on p?). (ii)="" l:="" r?="" —="" r%,="" [="" mn="" ]="" +="" x11="" +="" 250,="" where="" i="" and="" arc="" fixed="" (but="" unknown)="" ta="" vectors="" in="" r2.="" 4.="" [10]="" there="" is="" a="" unique="" lincar="" mapping="" l:="" r*="" —="" r?="" such="" that="" 1="" —1="" 1="" d1]="" (2d="" 12)="" wee]="" d120)="" 0="" 1="" -="" —1="" find,="" with="" justification,="" the="" standard="" matrix="" of="" l.="" 5.="" [8]="" the="" matrix="" 2="" 21="" (3)="" 2="" 12="" 3="" 1="" 22="" is="" the="" standard="" matrix="" of="" the="" lincar="" mapping="" l:="" r*="" —="" r?="" given="" by="" reflection="" about="" a="" plane="" p="" in="" 3-space="" that="" passes="" through="" the="" origin.="" find="" a="" scalar="" equation="" for="" p="" (hint:="" what="" docs="" l="" do="" to="" the="" vectors="" on="">