1. TRUE or FALSE. 2. Find the complete solution set to the system:
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1 TRUE or FALSE (a A homogenous system with more variables than equations has a nonzero solution True (The number of pivots is going to be less than the number of columns and therefore there is a free variable which provides nonzero solutions (b If x is a solution to the system of equations Ax = b for some vector b then x is also a solution to the same system False (In fact whenever b is nonzero and x is a solution to Ax = b then A(x =Ax =b b (c If A has a zero column then the homogenous system Ax = has a nonzero solution True (The zero column will contain no pivot and therefore the corresponding variable is free and provides nonzero solutions (d The only linear transformation which is both one-to-one and onto is the identity map False (For example it is easy to see that the map T : R R which takes every vector v R to v is a linear transformation which is both one-to-one and onto Find the complete solution set to the system: x + + x + x 4 x 5 =6 x + + x +x 4 x 5 = x +x x 4 +6x 5 = Using the row echelon form of the matrix given above, we see that,x 5 are free and we have: x + + x + x 4 x 5 =6 x x 4 + x 5 = 6 x 4 x 5 =4 x 4 =4+x 5 x = 6+x 4 x 5 = 6+(4+x 5 x 5 =+x 5 x =6 x x 4 + x 5 =6 (/( + x 5 (4 + x 5 +x 5 =4/ x 5 So the general solution is of the form x =/ x 5 x =/+x 5 x 4 =4+x 5 for arbitrary and x 5
2 Find the reduced row echelon matrix which is row equivalent to 4 Suppose A = (a Find all the solutions to the system of equations Ax = 4 (b Express the vector as a linear combination of columns of A 4
3 5 In each of the following cases, determine whether the given vector is in the set spanned by the columns of the given matrix: (a with 4 4 (b with 5,, is linearly independent in R and 6 Determine if the set S = 4 explain why
4 7 Determine if each of the following functions is a linear transformation If it is the case find the matrix representing the transformation with respect to the standard bases x (a L : R R,withL = x Solution: Suppose x = scalar Then Also L (x + y =L x x x L (c x =L = cl(x x + y + y x + y = L(x+L(y and y = cx c cx y y y are vectors in R and c R is a = c cx = c ( x = ( + y (x + y =( x +( y y These show that L is linear Also we can see that L(e =,L(e = andl(e = for the standard vectors in R,whichshowsthatthestandardmatrixisthe matrix ( (( (b L : R R x,withl ( x x = Solution: L is not linear For example we can see that L(e + e = L(e = and L(e =,so which shows that L is not linear (c L : R R with L x x L(e + L(e L(e +L(e = Solution: L is not linear For example L(e = L(e ( But ( (d L : R R x x x with L = Solution: It is easy to check like part (a above to see that L is linear To find the standard matrix, we have L(e = =( e +e and L(e = = e +e So the standard matrix of L is (
5 8 For the following linear transformations, find the standard matrix and also determine if they are one-to-one or onto x (a T : R R with T = x Solution: It follows from the definition of T that T (e =T(e =andt(e = So the standard matrix of T is the matrixa T =( T is onto because for every b R, T b = b In other terms, A T x = b is consistent for every b R T is not one-to-one, since T (e =T( =,orinothertermsa T x =hasmorethan one solution (b T : R R with T (e =e + e and T (e = e + e Solution: The definition of T immediately shows that the standard matrix of T is A T = T is not onto because there is no vector x R so that T (x =e,orinotherterms A T x = e is inconsistent T is one-to-one Recall that to prove this one only needs to show that the homogenous system A T x = has only the trivial solution This can be seen by finding the row echelon form of the matrix has two pivots ( (c T : R R x x + x with T = x Solution: T (e = = e e and T (e = matrix of T is A T = ( = e e Sothestandard T is not onto, because T (x = e has no solutions, or in other terms A T x = e is inconsistent T is not one-to-one either, because T (e e =T( =, inothertermsa T x = has more than one solutions 9 Suppose the following vectors in R are given v =, v =, v = (a Determine if the set S = {v, v, v } is linearly independent Solution: Recall that to check that S is linearly independent, we need to check if the equation x v + v + x v = has a nonzero solution Equivalently we need to check
6 if the homogenous system of equations Ax = has a nonzero solution, where A = To show this system has no nonzero solution, it is enough to find the reduced row echelon form of A and see that it has exactly three pivots Hence S is linearly independent (b Determine if S spans R Solution: To prove S spans R,weneedtoshowthesystemoflinearequationsAx = b has a solution for every b R ButwesawthatA has three pivots and therefore every row contains a pivot This implies that Ax = b is consistent (c Express the vector as a linear combination of elements of the vectors in S Solution: We need to find a solution to the system of equations Ax = This shows / = v + v v / / /
7 Suppose T : R n R n is a linear transformation which is not onto Answer the following questions and explain your answers (a What is the size of the standard matrix for T? (b How many pivots does the standard matrix of T has? (c Can T be one-to-one?
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