OHSX XM511 Linear Algebra: Multiple Choice Exercises for Chapter 2

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1 OHSX XM5 Linear Algebra: Multiple Choice Exercises for Chapter. In the following, a set is given together with operations of addition and scalar multiplication. Which is not a vector space under the given operations? (a) The set of all triples of real numbers (x, y, z) with the operations (x, y, z) + (x, y, z ) = (x + x, y + y, z + z ) and k(x, y, z) = (,, ). (b) The set of all n-tuples of real numbers of the form (x, x,, x) with the standard operations on R n. [ ] a (c) The set of all matrices of the form, with matrix addition and scalar b multiplication. (d) The set of all real-valued functions f defined everywhere on the real line and such that f() =, with the usual operations of addition and scalar multiplication for real-valued functions.. What does it mean that a subset W of V is closed under vector addition? (a) For any two vectors u, v W, we have αu + βv W. (b) For any two vectors u, v V, we have αu + βv W. (c) For any two vectors u, v W, we have u + v W. (d) For any two vectors u, v V, we have u + v W. 3. Let W be a non-empty subset of V. Which of the following is not a necessary and sufficient condition for W to be a subspace? (a) W is closed under vector addition and scalar multiplication. (b) For any two vectors u, v W and any scalars α, β we have αu + βv W. (c) For any two vectors u, v W and any scalar α we have αu + v W. (d) For any two vectors u, v V, we have αu + βv W. 4. Which is a subspace of R 4? (a) {(x, y, z, w) x y = z} (b) {(x, y, z, w) x y = z w + } (c) {(x, y, z, w) xy = zw} (d) {(x, y, z, w) x + y z}

2 5. Which describes the collection of all dimensional subspaces of R 3? (a) There are infinitely many: Take any four scalars a, b, c, d (not all zero); the plane defined by ax + by + cz + d = is a subspace. (b) There are infinitely many: Take any non-zero vector v and an additional vector w which is not a scalar multiple of v; span{v, w} is a subspace. (c) There is one: the x-y plane. (d) There are three: the x-y plane, the x-z plane, and the y-z plane. 6. Let u, v, x, y be non-zero vectors in V and α, β, scalars. Which could be true? (a) αu + βv = span{u, v} (b) span{u, v} {x, y} (c) {x} {u, v} (d) {u} span{x, y} 7. Which does not lie in the space spanned by the following two functions? f(x) = cos x, g(x) = sin x (a) sin x (b) cos x (c) (d) 5 sin x 8. Which set of row vectors does not span R 3? (a) {[,, ], [,, ], [3,, ]} (b) {[,, 4], [,, 3], [3,, 5]} (c) {[3,, 4], [, 3, 5], [5,, 9], [, 4, ]} (d) {[3,, ], [5, 3, ], [3,, 3], [, 7, ]} 9. How might you start a proof showing that v,..., v n are linearly independent in V? (a) Suppose there are scalars c,..., c n such that c v + + c n v n =. (b) We will show that there are scalars c,..., c n such that c v + + c n v n =. (c) Suppose c = = c n =. Then c v + + c n v n =. (d) We first show that for every v V there are scalars c,..., c n such that v = c v + + c n v n.

3 . What does it mean that the vectors v,..., v k are linearly dependent? (a) There exists scalars c,..., c k such that c v + + c k v k =. (b) There exists scalars c,..., c k, which are not all zero, such that c v + +c k v k =. (c) There exists scalars c,..., c k, none of which are zero, such that c v + +c k v k =. (d) The above choices may sound different in theory, but are the same in practice.. Which collection of row vectors in R 4 is linearly dependent? (a) [3,, 4, ], [, 3, 5, ], [ 3, 7, 8, 3] (b) [4, 4,, ], [,, 6, 6], [ 5,, 5, 5] (c) [4, 4, 8, ], [,, 4, ], [6,,, ], [6, 3, 3, ] (d) [,,, ], [,,, ], [,,, ], [,,, ]. Let A be an m n matrix and suppose v,..., v k are linearly dependent vectors in R n. What additional conditions are necessary to prove that Av,..., Av k are linearly dependent? (a) None. (b) That Av i for i =,..., k. (c) That A is a square matrix. (d) Both (b) and (c). 3. Suppose a matrix A has linearly dependent rows. Which must be true? (a) The system of equations Ax = has infinitely many solutions. (b) The system of equations A t y = has infinitely many solutions. (c) There is a row that is a scalar multiple of another. (d) The columns of A are linearly dependent. 4. If v,..., v k are linearly independent vectors in R n, then there is an invertible matrix that has these as its first k columns. Which fact would be the key to proving this? (a) Every linearly independent set of vectors in R n can be enlarged to a basis. (b) There are infinitely many vectors in R n \ {v,..., v k }. (c) Every spanning set of R n can be reduced to a basis. (d) Every invertible matrix has linearly independent columns. 3

4 5. If v,..., v k are linearly independent vectors in R n and A is an invertible matrix, then Av,..., Av k are linearly independent. The following is a proof of this statement. Which correctly completes the argument (in order)? Suppose there are scalars x,..., x k such that x Av + +x k Av k =. Then A(x v + + x k v k ) =. Because, it follows that x v + + x k v k =. Because, it follows that x = = x k =. This proves that Av,..., Av k are linearly independent. (a) No assumption is needed for the implication to be true; No assumption is needed for the implication to be true. (b) A is invertible; No assumption is needed for the implication to be true. (c) No assumption is needed for the implication to be true; v,..., v k are linearly independent (d) A is invertible; v,..., v k are linearly independent. 6. If span{v,..., v m } = span{v,..., v m, v m+ } in V, then which is true? (a) The vectors v,..., v m+ are linearly dependent. (b) The vectors v,..., v m are linearly independent. (c) dim(v ) m. (d) span{v,..., v m } = span{v,..., v m+ }. 7. Assume that v i for all i. Which is not true? (a) dim(span{v,..., v n }) n. (b) dim(span{x, v,..., v n }) dim(span{v,..., v n }) +. (c) If x / {v,..., v n }, then dim(span{x, v,..., v n }) > dim(span{v,..., v n }). (d) If x / span{v,..., v n }, then dim(span{x, v,..., v n }) = dim(span{v,..., v n }) Which is not a valid argument? (a) Since v,..., v n span V, it follows that dim(v ) n. (b) Since v,..., v n span V and since dim(v ) = n, these vectors form a basis. (c) Since v,..., v n are linearly independent in V, it follows that dim(v ) n. (d) Since v,..., v n are linearly independent and since dim(v ) = n, these vectors form a basis. 4

5 9. Which is a basis for the column space of the following matrix? 3 4 (a),, (b),, (c),, (d),,.. For the matrix below, let c,..., c 4 denote the column vectors in order. Which is not a basis for the column space? (a) c, c, c 3 (b) c, c, c 4 (c) c, c 3, c 4 (d) c, c 3, c 4. Suppose A is a 3 matrix. What is the maximal possible rank of A? (a) 3 (b) (c) 3 (d) 3. Suppose A is a 3 5 matrix. Which is impossible? (a) A has rank. (b) There is a 5 3 matrix B such that AB = I. (c) A has four linearly independent columns. (d) The row space of A has dimension. 5

6 3. If A has rank k, then it has a square submatrix of rank k. We begin a proof of this as follows: If a matrix A has rank k, then it has k linearly independent rows. Consider the submatrix A with these as its rows. Which ends the proof correctly? (a) The submatrix A is a square submatrix of A of rank k. (b) Choose the first k columns from the submatrix A ; these form a square submatrix of A of rank k. (c) Choose any k columns from the submatrix A ; these form a square submatrix of A of rank k. (d) Since the column rank of any matrix equals its row rank, there are k columns of A that are linearly independent. These form a square submatrix of A of rank k. 4. The following is a proof of a theorem. What does it prove? Let B be an n n submatrix of A. Since r(b) n, in the case n k, we have r(b) k, so the desired inequality is satisfied. Now consider the case where n > k. We use a proof by contradiction: Suppose r(b) > k. For convenience of notation, let m = r(b). Then, it follows from the definition of rank that B has m linearly independent rows, say r,..., r m. Consider the rows r,..., r m of A which originally contained these rows. Notice that if c r + +c mr m = for some scalars c,..., c m, then c r + +c m r m =. Since r,..., r m are linearly independent, we must have c = = c m =. Therefore r,..., r m are linearly independent. This implies that r(a) m > k, a contradiction. (a) If a matrix A has rank r(a) k, then any square submatrix B has rank r(b) k. (b) If a matrix A has rank r(a) k, then any square submatrix B has rank r(b) > k. (c) If a matrix A has rank r(a) k, then any square submatrix B has rank r(b) n. (d) If a matrix A has rank r(a) k, then any square submatrix B has rank r(b) n. 5. Which is false? (a) If A is an invertible n n matrix and B a square matrix of the same size, then r(ab) = r(b) = r(ba). (b) The column space of AB is contained in the column space of A. (c) The row space of AB is contained in the row space of A. (d) The rank of AB satisfies r(ab) min{r(a), r(b)}. 6. Let A be an n n matrix and B an n k matrix where k < n. Suppose AB =

7 Which must be true? (a) A is invertible. (b) The first k columns of A are linearly independent. (c) The column space of A has dimension k. (d) A has at least k linearly independent columns. 7. Consider the following in R m. Which procedure contains a faulty argument? (a) To find a basis for span{a,..., a n }, we make a matrix A with a,..., a n as its rows. We conclude that the non-zero rows of the row-reduced form of A is a basis. (b) To check whether a n+ is contained in span{a,..., a n } we make a matrix A with a,..., a n as its columns and the augmented matrix A = [A a n+ ]. As the ranks satisfy r(a) = r(a ), we conclude that a n+ is in the span. (c) To check whether a,..., a n are linearly independent, we make a matrix A with a,..., a n as its columns. As the row-reduced form of A has less than n non-zero rows, we conclude that the vectors are linearly dependent. (d) To check whether a,..., a n are linearly independent, we make a matrix A with a,..., a n as its columns. As the row-reduced form of A has a zero row, we conclude that the vectors are linearly dependent. 8. Suppose a sequence of elementary row operations are performed on a matrix. Which might change? (a) Its row space. (b) The dimension of its row space. (c) Its column space. (d) The dimension of its column space. 9. Suppose the solutions to a system Ax = b of equations form a k dimensional subspace of R n. Which does not follow? (a) The matrix A has n columns. (b) The column vector b =. (c) The rank r([a b]) = k. (d) The row-reduced form of A will have n k non-zero rows. 7