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1 Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are mutliples of the first. And the fifth column is times the third column minus times the first. So the second, third, and fifth columns are redundant. And a basis for the image is just and. To get a basis for the kernel we look at Ax. This tells us that relationship between entries of x are just x 5x x 4 +x 5 and x 4 x 5. So this gives us a basis of elements: x x x 5 x x x 5 5 x x x 5.. Find the reduced rowechelon form of the given matrix A. Then find a basis of the image of A and a basis for the kernel of A. 4 8 A There are leading ones in thefirst two columns of rref(a), So a basis for 4 the image is 4 and 5. We know that x is in the kernel is equivalent 7 9 to Ax. That s equivalent to rref(a)x. Which is equivalent
2 to x 6x and x 5x. Which is equivalent to x x 6 5 is a basis for the kernel So.. A subspace V of R n is called a hyperplane if V is defined by the homogeneous equation c x + c x c n x n where at least one of the coefficients c i is nonzero. What is the dimension of a hyperplane in R n? Justify your answer carefully. What is a hyperplane in R? What is it in R. Define C c c... c n. Notice that v is in V exactly when Cv. So ker(c) V. Call the first nonzero entry of C by k. Then rref(c) k C. So rref(c) has one leading nonzero, i.e. rank(c). By the RankNullity Theorem, dim(ker(c)) + rk(c) n. So dim(v ) dim(ker(c) n rk(c) n. So in R, a hyperplane is dimensional, or a plane. In R, a hyperplane is dimensional it s a line...9 We are told that a certain 5 5 matrix A can be written as A BC where B is 5 4 and C is 4 5. Explain how you know that A is not invertible. Since C is 4 5, we know that im(c) is a subspace of R 4. So dim(im(c)) 4. By the RankNullity theorem, we know dim(ker(c)) + dim(im(c)) 5. This means dim(ker(c)). So there is some vector v in the kernel of C. Now for any 5 5 matrix E at all, EAv EBCv EB v. So no E can make it true that EA I 5. In other words, A is not invertible...5 Find a basis for the row space of the matrix A Row reducing does not change the row space. So we want to row reduce A:
3 The two remaining nonzero rows arenot multiples of each other. So a basis for the row space is and...56 An n n matrix A is called nilpotent if A m for some positive integer m. Examples are triangular matrices whose entries on the diagonal are all. Consider a nilpotent n n matrix A and choose the small number m such that A m. Pick a vector v in R n such that A m v. Show that the vectors v, Av, A v,..., A m v are linearly independent. Suppose that c v + c Av + c A v c m A m v If all the c s before c i were, we will show that c i also. This means that all the c s must be. Consider the situation where all the c s before c i are. Then we get: c i A i v + c i+ A i+ v c m A m v If we multiply this equation by A m i we get: c i A m v + c i+ A m v c m A m i v c i A m v + A m ( c i+ v + c i+ Av c m A m i v ) c i A m v + ( c i+ v + c i+ Av c m A m i v ) c i A m v But we picked v so that A m v. Thus c i must also be, which results in all the c s being. So the vectors v, Av, A v,..., A m v are linearly independent..4. Find the matrix of the linear transformation T (x) Ax with respect to the basis B (v, v ). A ; v, v B S AS.4.6 Find the matrix of the linear transformation T (x) Ax with respect to the basis B (v,..., v m ). A ; v 4, v
4 B S AS Find a basis B of R n such that the Bmatrix B of the given linear transformation T is diagonal. T is the reflection about the line in R spanned by. We want to pick vectors v so that T (v) c v v for some c v. that way, the offdiagonal ( ) entries of B will be zero. Vectors on the line don t move, so T. Also, if we pick something perpendicular to the ( ) line, its reflection is just its negation. So T ( ). So according to the basis we will get B which is of the form we want. Ch. Ex. True or False? B, The span of vectors v, v,..., v n consists of all linear combinations of vectors v, v,.... True. This is the definition of span. 7 If u + v + 4w 5u + 6v + 7w, then the vectors u, v, w must be linearly dependent. True. w u v, so w is redundant. If A is a 5 6 matrix of rank 4, then the nullity of A is. False. We should get rank plus nullity equalling 6, not 5. a 8 The vectors of the form b (where a and b are arbitrary real a numbers) form a subspace of R 4. True. This set includes and is closed under both addition and multiplication. 4
5 4.. Is the following subset of P a subsapce? Find a basis for it if it is. {p(t) p() } Yes. () so is included. If p () and p () then (p + p )() p () + p () + so it is closed under addition. And if p() then (rp)() rp() r, so it is closed under scalar multiplication. Since dim (P ) and is not in the subspace, we know the dimension of the subspace must be at most. Since x and x 4 are in the subspace and linearly independent, they must be a basis Is this subset of R a subspace? The invertible matrices No. This set actually fails all three requirements of a subspace. It does not have. It has both I and I, but it doesn t I + ( I ), so it isn t closed under addition. It has I but it does not have I, so it is not closed under multiplication either. a b 4.. Find a basis for the space of all matrices A in R c d such that a d. If A is as written above, then A a + b + c As those three matrices are linearly independent, we get a basis of,, So the dimension is If B is a diagonal 4 4 matrix, what are the possible dimensions of the space of all 4 4 matrices A that commute with B. b B b b b 4 We need to have that (AB) ij (BA) ij. This tells us that A ij b j b i A ij. This tells us (b j b i )A ij. I.e. b i b j or A ij. So we can make a basis out of matrices E ij which have a in the ijth position and elsewhere, only including the E ij s for which b i b j. If all the b s are the same, we include all 6 E s. If of the b s are the same, we include the 9 E s for those along with the E for the remaining B with itself. This totals to. If there are two pairs of b s, then inside for each pair we have 4 E s. If there is one pair, and two other values, we get 4 E s for the pair plus one for each of the others, totalling 6. And finally, if all for b s are different, we get one E value for each, for 4 total. So the possible dimensions of this space are 4, 6, or 6. 5
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