Test Review Problems Spring 25 Let T HomV and let S be a subspace of V Define a map τ : V /S V /S by τv + S T v + S Is τ well-defined? If so when is it well-defined? If τ is well-defined is it a homomorphism? Determine imτ and kerτ For τ to be well-defined if u + S v + S we must have τu + S τv + S By the definition of τ we must have T u + S T v + S but T u + S T v + S T u T v S T u v S Additionally u + S v + S u v S So τ is well-defined if u v S T u v S which means that S must be T -invariant To show that τ is a homomorphism we need only show that τa[u] + b[v] aτ[u] + bτ[v] which follows from the linearity of T and by coset addition/scalar multiplication Suppose [u] u + S V /S is in kerτ so τu + S S Then we must have T u + S S T u S So kerτ S For w + S V /S to be in imτ there must be a v V such that T v w in which case we have τv + S T v + S w + S Thus imτ imt + S 2 Let v V be a nonzero vector in the vector space V Show that there exists an f HomV F such that f v Prove the contrapositive statement: if f v for every f HomV F then v Prove that T HomV W is an isomorphism if and only if T maps a basis of V to a basis of W a Assume T is bijective Suppose n c i T α i where α {α α n } is a basis i of V By linearity we then have T n c i α i which implies n c i α i since kert {} Hence c i for i n since α is a basis b Assume α {α α n } and T α{t α T α n } are bases for V and W respectively i i
Test Review Problems Spring 25 First we show that T is injective Suppose T v so T n c i α i By linearity we have n c i T α i i which implies c i for i n since T α is linearly independent Hence v kert {} Now since T α is a basis of W we have LT α W Let w W have expression w n a i T α i Again appealing to the linearity of T we have i n n a i T α i T a i α i T v i for some v V So T is surjective 4 Let T HomV where V is a vector space A subspace S of V is called T -invariant if T S S Assume S is a one-dimensional T -invariant subspace of V Show that S consists of scalar multiples of an eigenvector of T This shows that an eigenvector of T is the basis of a one-dimensional T -invariant subspace of V In class we showed that any one-dimensional subspace generated by an eigenvector of T is T -invariant Now we show that any one-dimensional T -invariant subspace must be generated by an eigenvector of T Suppose M {av v V } is such that T M M for some fixed vector v This means that if av M then T av M so T av bv for some ab F Since we are working over a field we may divide scalars hence so v is an eigenvector of T T v b a v 5 Let T HomV where dimv < Assume rankt 2 rankt and show that imt kert {} 6 Determine a projection P which projects R 2 onto the subspace spanned by T along the subspace spanned by 2 T i i 2
Test Review Problems Spring 25 If uv t R 2 is such that uv t a t + b2 t then P uv t a t The expression of uv t in the basis { t 2 t } is /2u v 2 t + /u + v2 t So for an arbitrary vector in R 2 we should have P uv t /2u v t u2/ 2/ t + v // t hence P 2/ / 2/ / 7 Let V be a finite-dimensional vector space and let S S m be subspaces of V such that V S + + S m and dimv dims + + dims m Show that V S S m 8 Prove that if P M HomV is the projection of M along N where V M N then I V P M is the projection of N along M If v v m + v n then P M v v m so I v P M v v m + v n v n v n which is enough to show that I V P M is the projection of N along M 9 Let T HomV such that every subspace of V is T -invariant Show that T is a scalar multiple of the identity operator Think about how T must act on subspaces generated by a single vector then on subspaces generated by two vectors etc In particular look at how T acts on Lα where α is a basis of V This is similar to number 4 Recall that if S is a subspace of V the codimension of S is codims dimv /S Suppose S and S 2 are two subspaces of V both having finite codimension a Show that S S 2 has finite codimension b Show that codims S 2 codims + codims 2 c Now suppose codims codims 2 Show that dims /S S 2 dims 2 /S S 2 Let f HomV F be nonzero so f v for at least some v V Show that V / kerf F By the first isomorphism theorem we know imf V / kerf so what must be shown is that imf F ie f is surjective Suppose f v a F where a By linearity for any b F f b a v b af v b
Test Review Problems Spring 25 2 Let T HomV and S a subspace of V such that T α α for every α S a Show that T induces a homomorphism τ HomV /S V /S b If τ is the identity map on V /S show that the map P T I V satisfies P 2 a Define τ by τv + S T v + S b Let δ {δ δ 2 δ } be the standard basis of R Let T HomR R be defined by 2 T δ 2 ; T δ 2 ; T δ 2 Let α {α α 2 α } be another basis of R where α 2 T α 2 T and α T Compute the two matrix representations Γ δδt and Γ ααt 4 Consider the differential equation d2 y yt Show the solutions of this dt2 differential equation form a vector space Determine a basis of this vector space: a if you only consider real solutions b if you allow for complex solutions Using the transformation yt e rt yields the equation r 2 + r ±i In the real case solutions are of the form yt Acost + Bsint so {costsint} is a basis In the complex case a basis is {e it e it } 5 Solve the linear system 6 2 6 u v over the field Z p for p 57 If p 7 how many solutions are there? We may row-reduce to determine solutions as usual but we must use modular arithmetic In Z 5 the orignal matrix is equivalent to 2 2 4
Test Review Problems Spring 25 We may also explicitly find the inverse matrix The determinant of this matrix in Z 5 is 2 mod 5 The multiplicative inverse of 2 is Also 2 mod 5 so the inverse matrix is 2 2 4 4 Thus u v 4 4 5 mod 5 A similar procedure yields the solution 6 t in Z In Z 7 the determinant is 28 mod 7 If we row-reduce we find that there are infinitely many solutions: { } 4 4 + b b Z 6 Determine a basis of V {A M n R A t A} 7 Let W R 4 be the space of solutions of Ax where 2 2 A Determine a basis of W Via row reduction we have 2 2 which yields the equations 4 x x + x 4 x 2 + 4x x 4 We may freely set x ax 4 b which yields the family of solutions 4 a + b ab R 5
Test Review Problems Spring 25 Thus a basis of W is 4 8 For Exercise in the lecture notes prove that M n F S T where S is the subspace of symmetric matrices and T is the subspace of skew-symmetric matrices 9 Suppose T HomV where V is a vector space Define V λ {v V T v λv} to be the subspace of eigenvectors of T with eigenvalue λ Show that V λ is T -invariant Identical to part of Problem 4 2 If S S 2 are both T -invariant show S + S 2 and S S 2 are T -invariant 2 Let A M mn R Prove that the vector space of solutions of Ax has dimension at least n m Note that A is the matrix of some homomorphism T mapping an n-dimensional vector space into an m-dimensional vector space We are looking for the dimension of the kernel of T ie the nullity By the rank-nullity theorem we know dimv rankt + nullityt What is the maximum dimension of rankt in this case? 22 Let MN be subspaces of a vector space V and define operations u v + u 2 v 2 u + u 2 v + v + 2 cuv cucv where uu u 2 M vv v 2 N and c F Show that this makes the product space M N into a vector space Define a map T : M N V by T uv u + v Show that T is a homomorphism 2 Consider the vector space M n F of n n matrices with entries in F Define the trace endomorphism denoted by tr by tra n a ii i where A a ij n ij That is the trace endomorphism returns the sum of the diagonal entries of a matrix or transformation in general Define the space 6
Test Review Problems Spring 25 slnf {A M n F tra } This is an example of a particular type of algebra but we may simply regard it as a vector space for this example Determine a basis of sl2f and slf Do you recognize a pattern? See if you can generalize this to give a basis of slnf 24 Let T HomV W Show that ker T {} if and only if T maps each linearly independent subset of V onto a linearly independent subset of W Half of Problem 25 Let T be the homomorphism from R to R 2 defined by T uvw t u + v2w u t a If B and B 2 are the standard bases for R and R 2 respectively determine the matrix of T relative to B B 2 b Let B B 2 { be bases for R and R 2 respectively Compute the matrix of T relative to B B 2 26 Let T be the homomorphism from R 2 to R 2 defined by T uv t vu t a Determine the matrix of T relative to the standard basis of R 2 b Let B { 2 be a basis for R 2 Compute the matrix of T relative to B } } 7