Math 313 Chapter 5 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 5.1 Real Vector Spaces 2 2 5.2 Subspaces 3 3 5.3 Linear Independence 4 4 5.4 Basis and Dimension 5 5 5.5 Row Space, Column Space, and Nullspace 6 6 5.6 Rank and Nullity 7 1
1 5.1 Real Vector Spaces Important Formulas to Remember Vector Space Axioms (1) If u and v are objects in V, then u + v is in V. (2) u + v = v + u (3) u + ( v + w) = ( u + v) + w (4) There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u for all u in V (5) For each u in V, there is an object u in V, called a negative of u such that u + ( u) = ( u) + u = 0 (6) If k is any scalar and u is any object in V, then k u is in V. (7) k( u + v) = k u + k v (8) (k + m) u = k u + m u (9) k(m u) = (km) u (10) 1 u = u Determine whether or not the set of all n-tuples of real numbers of the form (x, x,..., x) with the standard operations on R n is a vector space. Determine whether the set of all pairs of real numbers (x, y) with the operators (x, y) + (x, y ) = (x + x + 1, y + y + 1), k(x, y) = (kx, ky) forms a vector space. Determine whether the set of all 2 x 2 matrices of the form [ a ] a + b a + b b with matrix addition and scalar multiplication forms a vector space. Show that the set of all triples of real numbers with the standard vector addition but with scalar multiplication defined by k(x, y, z) = (k 2 x, k 2 y, k 2 z) fails to be a vector space by identifying the axiom(s) it fails to hold. Is it possible to have a vector space with exactly two distinct vectors in it? Explain your reasoning. 2
2 5.2 Subspaces Important Formulas to Remember Closure Axioms (1) If u and v are vectors in W, then u + v is in W (2) If k is any scalar and u is any vector in W, then k u is in W Determine whether all matrices of the form [ ] a b 0 c is a subspace of M 22 Determine whether or not the set of all nxn matrices A such that tr(a) = 0 is a subspace of M nn. Show that the solution vectors of a consistent nonhomogeneous system of m linear equations in n unknowns do not form a subspace of R n. Under what conditions will two vectors in R 3 span a plane? A line? Under what conditions will it be true that span{ u} =span{ v}? Explain. 3
3 5.3 Linear Independence Linear Independence If the only solution to the vector equation k 1 v 1 + k 2 v 2 +... + k 2 v r = 0 is k 1 = k 2 =... = k r = 0, then the set { v 1, v 2,... v r } is linearly independent. Theorem A finite set of vectors that contains the zero vector is linearly dependent. f 1 (x) f 2 (x)... f n (x) f 1 Wronskian W (x) = (x) f 2 (x)... f n(x)... f (n 1) 1 (x) f (n 1) 2 (x)... f n (n 1) (x) The vectors are independent if the result is not identically equal to zero. If it is zero, then the test yields no information. Explain why the following are linearly dependent sets of vectors. (Solve by inspection) (a) u 1 = ( 1, 2, 4) and u 2 = (5, 10, 20) (b) p 1 = 3 2x + x 2 and p 2 = 6 4x + 2x 2 Which of the following sets of vectors in P 2 are linearly dependent? (a) 3 + x + x 2, 2 x + 5x 2, 4 3x 2 (b) 1 + 3x + 3x 2, x + 4x 2, 5 + 6x + 3x 2, 7 + 2x x 2 Show that the vectors v 1 = (1, 2, 3, 4), v 2 = (0, 1, 0, 1), and v 3 = (1, 3, 3, 3), form a linearly dependent set in R 4. Show that every set with more than three vectors from P 2 is linearly dependent. Prove: The space spanned by two vectors in R 3 is a line through the origin, a plane through the origin, or the origin itself. Determine whether the Wronskian tells you if the following sets of vectors in C 3 (R) are linearly independent. If the Wronskian is inconclusive, say so. (a) {1, sin x, cos x} (b) {x, 3x + 2, 9x 5} 4
4 5.4 Basis and Dimension Basis A set of vectors is a basis for a vector space if the set it linearly independent and it spans the space. Uniqueness of If a set of vectors is a basis for a vector space then Basis Representation every vector can be expressed in the form v = c 1 v 1 + c 2 v 2 +... + c n v n in exactly one way. Dimension The number of vectors in the basis of a vector space. Which of the following sets of vectors are bases for R 3? (a) (3, 1, 4), (2, 5, 6), (1, 4, 8) (b) (2, 3, 1), (4, 1, 1), (0, 7, 1) Which of the following sets of vectors are bases for P 2? (a) 1 + x + x 2, x + x 2, x 2 (b) 4 + x + 3x 2, 6 + 5x + 2x 2, 8 + 4x + x 2 Find the coordinate vector of w relative to the basis S = { u 1, u 2 } for R 2. (a) u 1 = (2, 4), u 2 = (3, 8); w = (1, 1) (b) u 1 = (1, 1), u 2 = (0, 2); w = (a, b) Determine the dimension of and a basis for the solution space of the system x + y + z = 0 3x + 2y 2z = 0 4x + 3y z = 0 6x + 5y + z = 0 Determine the basis for the subspace of all vectors of the form (a, b, c) where a = 2c + b.. 5
5 5.5 Row Space, Column Space, and Nullspace Row Space The subspace spanned by the row vectors of a matrix. Column Space The subspace spanned by the column vectors of a matrix. Nullspace The solution space of the homogeneous system of equations. Theorem Elementary row operations do not affect the fundamental subspaces. Express the product A x as a linear combination of the column vectors of A. 3 6 2 5 4 0 1 2 3 1 2 5 1 8 3 Determine whether b is in the column space of A, and if so, express b as a linear combination of the column vectors of A. 1 1 1 5 A = 9 3 1 b = 1 1 1 1 1 Find the fundamental subspaces of A 1 4 5 2 (a) 2 1 3 0 1 3 2 2 1 3 2 2 1 0 3 6 0 3 (b) 2 3 2 4 4 3 6 0 6 5 2 9 2 4 5 Prove that row vectors of an nxn invertible matrix A form a basis for R n. 6
6 5.6 Rank and Nullity Theorem If A is any matrix, then the row space and column space of A have the same dimension. Rank rank(a)= the dimension of the row space/column space. Nullity nullity(a) = the dimension of the nullspace. Dimension Theorem rank(a) + nullity(a) = n Equivalent Statements If A is an nxn matrix, then the following are equivalent: to Date (a) A is invertible (b) A x = 0 has only the trivial solution. (c) The reduced row-echelon form of A is I n (d) A is expressible as a product of elementary matrices. (e) A x = b is consistent for every nx1 matrix b (f) A x = b has exactly one solution for every nx1 matrix b (g) det(a) 0. (h) The range of T A is R n. (i) T A is one-to-one. (j) The column vectors of A are linearly independent. (k) The row vectors of A are linearly independent. (l) The column vectors of A span (R) n. (m) The row vectors of A span (R) n. (n) The column vectors of A form a basis for (R) n. (o) The row vectors of A form a basis for (R) n. (p) A has rank n. (q) A has nullity 0. Find the rank and nullity of the matrices; verify that the values obtained satisfy the dimension theorem. 1 4 5 2 2 1 3 0 1 3 2 2 1 3 2 2 1 0 3 6 0 3 2 3 2 4 4 3 6 0 6 5 2 9 2 4 5 Prove: If k 0, then A and ka have the same rank. Suppose that A is a 3 x 3 matrix whose nullspace is a line through the origin in 3-space. Can the row or column space of A also be a line through the origin? Explain. 7