Math 3013 Homework Set 1 1. Compute v + w, v − w and then draw coordinate axes and sketch, using your answers the vectors v, w, v + w, and v − w. (a) v = [2, −1], w = [−3, −2]. (b) v = i + 3j + 2k, i + 3j + 4k, where i = [1, 0, 0], j = [0, 1, 0], k = [0, 0, 1] are the standard basis vectors of R3 . 2. Let u = [1, 2, 1, 0], v = [−2, 0, 1, 6] and w = [3, −5, 1, −2]. Compute u − 2v + 4w. 3. Find the vector which, when translated, represents geometrically an arrow reaching from the point (−1, 3) to the point (4, 2) in R2 . 4. Let u = [−1, 3, 4] and v = [2, 1, −1]. Compute k−uk and kv + uk. 5. Compute the angle between [1, −1, 2, 3, 0, 4] and [7, 0, 1, 3, 2, 4] in R6 . 6. Prove that (2, 0, 4), (4, 1, −1) and (6, 7, 7) are the vertices of a right triangle in R3 . 7. Classify the vectors as parallel, perpendicular, or neither. If they are parallel, state whether they have the same or opposite directions. (a) [−1, 4] and [8, 2]. (b) [3, 2, 1] and [−9, −6, −3]. 8. Find the distance between the points (2, −1, 3) and (4, 1, −2) in R3 . 9. Let A= −2 4 1 3 0 −1 , B= 4 5 1 −1 −2 3 , 2 C = 0 −3 (a) 3A (b)A + B (c) AB (d) A2 (e) (2A − B) D (f) ADB 10. Consider the row and column vectors x = [−2, 3, −1] , Compute the matrix products xy and yx. 11. Mark the following statements True or False. 1 4 y = −1 3 −1 6 2 , −4 D = 3 −1 2 5 −3 2 a. If A = B, then AC = BC. b. If AC = BC, then A = B. c. If AB = 0, then A = 0 or B = 0. d. If A + C = B + C, then A = B. e. If A2 = I, then A = ±I. f. If B = A2 and if A is an n × n matrix and symmetric, then bii ≥ 0 for i = 1, 2. . . . , n. g. If AB = C and if two of the matrices are square, then so is the third. h. If AB = C and if C is a column vector then so is B. i. If A2 = I, then An = I for all integers n ≥ 2. j. If A2 = I, then An = I for all even integers n ≥ 2. 3 Math 3013 Homework Set 2 1. Reduce the following matrices to row-echelon form, and reduced row-echelon form. (a) 2 1 3 4 2 6 1 3 −1 (b) 0 −1 1 1 2 −1 1 2 1 −3 5 5 3 0 3 9 (c) −1 1 2 0 0 1 4 0 0 −1 2 4 0 1 3 −4 3 −3 −6 0 2. Describe all the solutions of a linear system whose corresponding augmented matrix can be row reduced to the given matrix. 1 0 2 0 1 (a) 0 1 1 3 −2 0 0 0 0 0 1 0 (b) 0 0 −1 0 0 0 2 0 3 1 0 1 4 2 0 0 0 −1 0 0 0 0 3. Find all solutions of the given linear systems. (a) 2x − y = 8 6x − 5y = 32 y+z = 6 (b) 3x − y + z = −7 x + y − 3z = −13 4. Determine whether the vector 3 b= 5 3 4 is in the span of the vectors 0 v1 = 2 4 , 1 v2 = 4 −2 , −3 v3 = −1 5 5 Math 3013 Homework Set 3 1. Find the inverses of the following matrices. If a matrix inverse exists, express it as a product of elementary matrices. 1 0 1 1 3 4 6 8 1 (c) 0 0 0 1 0 1 2 −1 (a) (b) 2 (d) 3 0 1 1 −1 4 5 1 2. Find the inverse of 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 5 3. Determine whether the span of the column vectors 1 0 0 −1 1 0 −3 0 of the given matrix is R4 . 1 −1 −3 4 −1 2 0 −1 4. Show that the following matrix is invertible and find its inverse. 2 −3 A= 5 −7 5. Given that A−1 1 = 0 4 2 3 1 1 1 2 If possible, find a matrix C such that 1 AC = 0 4 2 1 1 6 Math 3013 Homework Set 4 1. Determine whether the indicated subset is a subspace of the given Rn . (a) W = {[r, −r] | r ∈ R} in R2 (b) W = {[n, m] | n and n are integers} in R2 (c) W = {[x, y, z] | x, y, z ∈ R and z = 3x + 2} in R3 (d) W = {[x, y, z] | x, y, z ∈ R and z = 1, y = 2x} in R3 (e) W = {[2x1 , 3x2 , 4x3 , 5x4 ] | xi ∈ R} in R4 2. Prove that the line y = mx is a subspace of R2 . (Hint: write the line as W = {[x, mx] | x ∈ R}.) 3. Find a basis for the solution set of the following homogeneous linear systems. (a) 3x1 + x2 + x3 = 0 6x1 + 2x2 + 2x3 = 0 −9x1 − 3x2 − 3x3 = 0 (b) 2x1 + x2 + x3 + x4 = 0 x1 − 6x2 + x3 = 0 3x1 − 5x2 + 2x3 + x4 = 0 5x1 − 4x2 + 3x3 + 2x4 = 0 (c) x1 − x2 + 6x3 + x4 − x5 = 0 3x1 + 2x2 − 3x3 + 2x4 + 5x5 = 0 4x1 + 2x2 − x3 + 3x4 − x5 = 0 3x1 − 2x2 + 14x3 + x4 − 8x5 = 0 2x1 − x2 + 8x3 + 2x4 − 7x5 = 0 4. Solve the following linear systems and express the solution set in a form that illustrates Theorem 1.18. (a) 2x1 − x2 + 3x3 = −3 4x1 + 4x2 − x4 = 1 7 (b) 2x1 + x2 + 3x3 = 5 x1 − x2 + 2x3 + x4 = 0 4x1 − x2 + 7x3 + 2x4 = 5 −x1 − 2x2 − x3 + x4 = −5 5. Mark each of the following statements True or False. a. A linear system with fewer equations than unknowns has an infinite number of solutions. b. A consistent linear system with fewer equations than unknowns has an infinite number of solutions. c. If a square linear system Ax = b has a solution for every choice of column vector b, then the solution is unique for each choice of b. d. If a square system Ax = 0 has only the trivial solution x = 0, then Ax = b has a unique solution for every column vector b with the appropriate number of components. e. If a linear system Ax = 0 has only the trivial solution x = 0, then Ax = b has a unique solution for every column vector b with the appropriate number of components. f. The sum of two solution vectors of any linear system is also a solution vector of the system. g. The sum of two solution vectors of any homogeneous linear system is also a solution vector of the system. h. A scalar multiple of a solution vector of any homogeneous linear system is also a solution vector of the system. i. Every line in R2 is a subspace of R2 generated by a single vector. j. Every line in R2 through the origin in R2 is a subspace of R2 generated by a single vector. 8 Math 3013 Homework Set 5 1. Give a geometric criterion for a set of two distinct nonzero vectors in R2 to be dependent. 2. Give a geometric criterion for a set of two distinct nonzero vectors in R3 to be dependent. 3. Find a basis for the subspace spanned by the vectors [1, 2, 1, 1], [2, 1, 0, −1], [−1, 4, 3, 8], [0, 3, 2, 5] ∈ R4 . 4. Find a basis for the column space of the matrix 2 5 A= 1 6 3 2 7 −2 1 1 2 0 5. Find a basis for the row space of the matrix 1 A= 2 3 3 0 2 5 4 8 7 2 7 6. Determine whether the following sets of vectors are dependent or independent. (a) {[1, 3] , [−2, −6]} in R2 . (b) {[1, −4, 3] , [3, −11, 2] , [1, −3, −4]} in R3 . 7. For each of the following matrices find the rank of the matrix, a basis for its row space, a basis for its column space, and a basis for its null space. (a) A= 2 0 −3 3 4 2 1 2 (b) 0 6 6 1 2 1 A= 4 1 −3 1 3 2 3 1 4 0 (c) 0 A= 2 0 1 1 2 8. Determine whether the following matrix is invertible, 0 −9 1 2 A= 4 1 1 3 2 0 1 1 2 1 by finding its rank. −9 2 1 1 −3 4 2 0 9 9. Determine whether the following statements are true or false. (a) The number of independent row vectors in a matrix is the same as the number of independent column vectors. (b) If H is a row-echelon form of a matrix A, then the nonzero column vectors of H form a basis for the column space of A. (c) If H is a row-echelon form of a matrix A, then the nonzero row vectors of H from a basis for the row space of A. (d) If an n × n matrix A is invertible then rank (A) = n. (e) For every matrix A we have rank(A) > 0. (f) For all positive integers m and n, the rank of an m × n matrix might be any number from 0 to the maximum of m and n. (g) For all positive integers m and n, the rank of an m × n matrix might be any number from 0 to the minimum of m and n. (h) For all positive integers m and n, the nullity of an m × n matrix might be any number from 0 to n. (i) For all positive integers m and n, the nullity of an m × n matrix might be any number from 0 to m. (j) For all positive integers m and n, with m ≥ n, the nullity of an m × n matrix might be any number from 0 to n. 10. Determine which of the following mappings are linear transformations. (a) T : R3 → R2 : T ([x1 , x2 , x3 ]) = [x1 + x2 , x1 − 3x2 ] (b) T : R3 → R4 : T ([x1 , x2 , x3 ]) = [0, 0, 0, 0] (c) T : R3 → R4 : T ([x1 , x2 , x3 ]) = [1, 1, 1, 1] (d) T : R2 → R3 : T ([x1 , x2 ]) = [x1 − x2 , x2 + 1, 3x1 − 2x2 ] 11. (Problems 2.3.5 and 2.3.7 in text). For each of the following, assume T is a linear transformation, from the data given, compute the specified value. (a) Given T ([1, 0]) = [3, −1], and T ([0, 1]) = [−2, 5], find T ([4, −6]). (b) Given T ([1, 0, 0]) = [3, 1, 2], T ([0, 1, 0]) = [2, −1, 4], and T ([0, 0, 1]) = [6, 0, 1], find T ([2, −5, 1]). 12. Find the standard matrix representations of the following linear transformations. (a) T ([x1 , x2 ]) = [x1 + x2 , x1 − 3x2 ] (b) T ([x1 , x2 , x3 ]) = [x1 + x2 + x3 , x1 + x2 , x1 ] 13. If T : R2 → R3 is defined by T ([x1, x2 ]) = [2x1 + x2 , x1 , x1 − x2 ] and T 0 : R3 → R2 is defined by T 0 ([x1 , x2 , x3 ]) = [x1 − x2 + x3 , x1 + x2 ], find the standard matrix representation for the linear transformation T 0 ◦ T that carries R2 into R2 . Find a formula for (T 0 ◦ T ) ([x1 , x2 ]). 10 14. Determine whether the following statements are true or false. (a) Every linear transformation is a function. (b) Every function mapping Rn to Rm is a linear transformation. (c) Composition of linear transformations corresponds to multiplication of their standard matrix representations. (d) Function composition is associative. (e) An invertible linear transformation mapping Rn to itself has a unique inverse. (f) The same matrix may be the standard matrix representation for several different linear transformations. (g) A linear transformation having an m × n matrix as its standard matrix representation maps Rn into Rm . (h) If T and T 0 are different linear transformations mapping Rn into Rm , then we may have T (ei ) = T 0 (ei ) for all standard basis vectors ei of Rn . (i) If T and T 0 are different linear transformations mapping Rn into Rm , then we may have T (ei ) = T 0 (ei ) for some standard basis vectors ei of Rn . (j) If B = {b1 , b2 , . . . , bn } is a basis for Rn and T and T 0 are linear transformations from Rn into Rm , then T (x) = T 0 (x) for all x ∈ Rn if and only if T (bi ) = T 0 (bi ) for i = 1, 2, . . . , n. 11 Math 3013 Homework Set 6 1. Determine whether the given set is closed under the usual operations of addition and scalar multiplication, and is a (real) vector space. (a) The set of all diagonal n × n matrices. (b) The set Pn of all polynomials in x, with real coefficients and of degree less than or equal to n, together with the zero polynomial. 2. Determine whether the following statements are true or false. (a) Matrix multiplication is a vector space operation on the set Mm×n of m × n matrices. (b) Matrix multiplication is a vector space operation on the set Mn×n of square n × n matrices. (c) Multiplication of any vector by the zero scalar always yields the zero vector. (d) Multiplication of a non-zero vector by a non-zero scalar always yields a non-zero vector. (e) No vector is its own additive inverse. (f) The zero vector is the only vector that is its own additive inverse. (g) Multiplication of two scalars is of no concern to the definition of a vector space. (h) One of the axioms for a vector space relates the addition of scalars, multiplication of a vector by scalars, and the addition of vectors. (i) Every vector spaces has at least two vectors. (j) Every vector space has at least one vector. 3. Determine whether the set of all functions f such that f (1) = 0 is a subspace of the vector space F of all functions mapping R into R. 4. Let P be the vector space of polynomials. Prove that span (1, x) = span (1 + 2x, x). 5. Determine whether the following set of vectors is dependent or independent in P : 1, 4x + 3, 3x − 4, x2 + 2, x − x2 6. Determine whether the following statements are true or false. (a) The set consisting of the zero vector is a subspace for every vector space. (b) Every vector space has at least two distinct subspaces. (c) Every vector space with a nonzero vector has at least two distinct subspaces. (d) If {v1 , v2 , . . . , vn } is a subset of a vector space then vi is in span (v1 , v2 , . . . , vn ) for i = 1, 2, . . . , n. (e) If {v1 , v2 , . . . , vn } is a subset of a vector space then vi + vj is in span (v1 , v2 , . . . , vn ) for all choices of i and j between 1 and n. (f) If u + v lies in a subspace W of a vector space V , then both u and v lie in W . 12 (g) Two subspaces of a vector space may have empty intersection. (h) If S = {v1 , v2 , . . . , vk } is independent, each vector in V can be expressed uniquely as a linear combination of vectors in S. (i) If S = {v1 , v2 , . . . , vn } is independent and generates V , then each vector in V can be expressed uniquely as a linear combination of vectors in S. (j) If each vector in V can be expressed uniquely as a linear combination of vectors in S = {v1 , . . . , vk }, then S is an independent set. 7. Let V be a vector space. Determine whether the following statements are true or false. (a) Every independent set of vectors in V is a basis for the subspace the vectors span. (b) If {v1 , v2 , . . . , vn } generates V , then each v ∈ V is a linear combination of vectors in this set. (c) If {v1 , v2 , . . . , vn } generates V , then each v ∈ V is a unique linear combination of vectors in this set. (d) If {v1 , v2 , . . . , vn } generates V and is independent, then each v ∈ V is a linear combination of vectors in this set. (e) If If {v1 , v2 , . . . , vn } generates V , then this set of vectors is independent. (f) If each vector in V is a unique linear combination of the vectors in the set {v1 , v2 , . . . , vn }, then this set is independent. (g) If each vector in V is a unique linear combination of the vectors in the set {v1 , v2 , . . . , vn }, then this set is a basis for V . (h) All vector spaces having a basis are finitely generated. (i) Every independent subset of a finitely generated vector space is a part of some basis for V . (j) Any two bases in a finite-dimensional vector space V have the same number of elements. 13 Math 3013 Homework Set 7 1. Find the coordinate vector of the given vector relative to the indicated ordered basis. (a) [−1, 1] ∈ R2 , relative to ([0, 1] , [1, 0]) (b) (c) (d) (e) [4, 6, 2] ∈ R3 , relative to ([2, 0, 0] , [0, 1, 1] , [0, 0, 1]) [3, 13, −1] ∈ R3 , relative to ([1, 3, −2] , [4, 1, 3] , [−1, 2, 0]) x3 + x2 − 2x + 4 in P3 , relative to 1, x2 , x, x3 x + x4 in P4 , relative to 1, 2x − 1, xn3 + x4 , 2x3 , x2 + 2 2 (f) x3 − 4x2 + 3x + 7 relative to B 0 = (x − 2)3 , (x − 2) , x − 2, 1 o 2. Let F be the vector space of functions f : R → R. Determine whetther the following functions T are linear transformations. (a) T : F → R : T (f ) = f (−4). (b) T : F → F : T (f ) = −f 3. Let C R xbe the space of all continuous functions mapping R to R, and let T : C → C be defined by T (f ) = 1 f (t)dt. If possible give three different functions in ker (T ). 4. Let V and V 0 be vector spaces having ordered bases B = (b1 , b2 , b3 ) and B 0 = (b01 , b02 , b03 , b04 ), respectively. Let T : V → V 0 be a linear transformation such that T (b1 ) = 3b01 + b02 + 4b03 + b04 T (b2 ) = b01 + 2b02 − b03 + 2b04 T (b3 ) = −2b01 − b02 + 2b03 Find the matrix representation AT of T relative to B, B 0 . 4. Let V and V 0 be vector spaces with ordered bases B = (b1 , b2 , b3 ) and B 0 = (b01 , b02 , b03 ), respectively. Let T : V → V 0 be a linear transformation such that T (b1 ) = b01 + 2b02 − 3b03 T (b2 ) = T (b3 ) = −2b01 − 3b02 − 4b03 3b01 + 5b02 + 2b03 (a) Find the matrix AT . (b) Use AT to find T (v)B 0 if vB = [2, −5, 1]. (c) Show that T is invertible, and find the matrix representation of T −1 relative to B 0 , B. 0 (d) Find T −1 (v0 )B if vB 0 = [−1, 1, 3]. −1 0 (e) Express T (b1 ), T −1 (b02 ), and T −1 (b03 ) as linear combinations of the vectors in B. 2 dp d p 5. Let T : P3 → P3 be the linear transformation defined by T (p) = dx2 − 4 dx + p. Find the matrix 2 3 representation AT of T using the ordered basis x, 1 + x, x + x , x of P3 . 14 Math 3013 Homework Set 8 1. Calculate the following determinants. 1 4 −2 0 (a) 5 13 2 −1 3 2 (b) 1 −2 1 (c) 0 1 2 (d) −1 2 −5 3 3 −2 1 0 −1 0 1 3 4 7 7 4 3 1 3 −4 2. Show by direct calculation that a1 b1 c1 a2 b2 c2 a3 b3 c3 a1 = − c1 b1 a2 c2 b2 a3 c3 b3 3. Mark each of the following True or False. (a) The determinant of a 2 × 2 matrix is a vector. (b) If two rows of a 3 × 3 matrix are interchanged, the sign of the determinant is changed. (c) The determinant of a 3 × 3 matrix is zero if two rows of the matrix are parallel vectors in R3 . (d) In order for the determinant of a 3 × 3 matrix to be zero, two rows must be parallel vectors in R3 . (e) The determinant of a 3 × 3 matrix is zero if the points in R3 lie in a plane. (f) The determinant of a 3 × 3 matrix is zero if the points in R3 lie in a plane through the origin. (g) The parallelogram in R2 determined by non-zero vectors a and b is a square if and only if a · b = 0. (h) The box in R3 determined by vectors a, b, and c is a cube if and only if a · b = a · c = b · c = 0 and a · a = b · b = c · c. (i) If the angle between two vectors a and b in R3 is π/4, then ka × bk = |a · b|. 2 (j) For any vector a in R3 we have ka × ak = kak . 4. Compute the determinants of the following matrices. 1 0 6 5. Find the cofactor of 5 for the matrix 4 1 −1 5 0 1 15 6. Suppose A is a 3 × 3 matrix such that det (A) = 2. Compute the following. (a) det A2 (b) det (3A) (c) det A−1 7. Mark each of the following True or False. (a) The determinant det (A) is defined for any matrix A. (b) The determinant det (A) is defined for any square matrix A. (c) The determinant of a square matrix is a scalar. (d) If matrix A is multiplied by a scalar c, then the determinant of the resulting matrix is c · det (A) . (e) If an n×n matrix A is multiplied by a scalar c, then the determinant of the resulting matrix is cn ·det (A). 2 (f) For every square matrix A, we have det AAT = det AT A = (det (A)) (g) If two rows and also two columns of square matrix are interchanged, the determinant changes sign. (h) The determinant of an elementary matrix is nonzero. (i) If det (A) = 2 and det (B) = 3, then det (A + B) = 5. (j) If det(A) = 2 and det (B) = 3, then det (AB) = 6. 8. Compute the determinants of the following matrices. 2 3 −1 1 (a) A = 5 −7 −3 2 −1 5 2 (b) A = 3 1 (c) A = (d) A = 2 3 0 0 0 0 0 0 5 −3 2 4 0 −3 −1 2 −4 3 7 −1 0 1 1 0 0 −1 2 0 4 1 −1 0 −3 2 0 0 −1 0 0 2 4 3 0 0 3 1 0 2 0 −3 −2 1 0 0 −3 2 0 0 4 0 0 0 16 (e) A = 2 0 −5 0 0 −1 3 0 0 1 4 0 0 2 6 0 0 0 0 1 4 0 0 −2 8 9. Use the corollary to Theorem 4.6 to find A−1 if A is invertible. 4 1 (a) A = 2 1 3 (b) A = −2 3 0 1 1 4 1 2 10. Solve the following systems of linear equations using Cramers’s Rule. (a) x1 − 2x2 3x1 + 4x2 (b) x1 + 2x2 − x3 2x1 + x2 + x3 3x1 − x2 + 5x3 = = 1 3 = −2 = 0 = 1 17 Math 3013 Homework Set 9 1. Find the characteristic polynomial, the real eigenvalues, and the corresponding eigenvectors for the following matrices. 7 5 (a) A = −10 −8 −7 16 1 1 (b) A = (c) A = −5 17 −2 2 0 0 2 −1 0 3 0 0 4 −5 0 9 0 0 2 −1 0 3 −1 (d) A = −4 4 1 (e) A = −8 8 −4 (f) A = −7 7 2. Find the eigenvalues λi and the corresponding eigenvectors vi for the following linear transformations. (a) T ([x, y]) = [2x − 3y, −3x + 2y] (b) T ([x1 , x2 , x3 ]) = [x1 + x3 , x2 , x1 + x3 ] 3. Mark each of the following statements True or False. (a) Every square matrix has real eigenvalues. (b) Every n × n matrix has n distinct (possible complex) eigenvalues. (c) Every n × n matrix has n not necessarily distinct and possibly complex eigenvalues. (d) There can be only one eigenvalue associated with an eigenvector of a linear transformation. (e) There can be only one eigenvector associated with an eigenvalue of a linear transformation. (f) If v is an eigenvector of a matrix A, then v is an eigenvector of A + cI for all scalars c. (g) If λ is an eigenvalue of a matrix A, then λ is an eigenvalue of A + cI for all scalars c. (h) If v is an eigenvector of an invertible matrix A, then cv is an eigenvector of A−1 for all non-zero scalars c. (i) Every vector in a vector space V is an eigenvector of the identity transformation of V into V . (j) Ever nonzero vector in a vector space V is an eigenvector of the identity transformation of V into V . 18 4. Find the eigenvalues λi , the corresponding eigenvectors vi of the following matrices. Also find an invertible matrix C and a diagonal matrix D such that D = C−1 AC. −3 4 (a) A = 4 3 3 1 7 −4 8 −5 6 (d) A = −2 16 3 −1 8 10 −6 7 −6 0 1 (b) A = (c) A = 2 4 −3 (e) A = 0 0 −3 2 −7 5. Determine whether or not the following matrices are diagonalizable. 1 2 6 (a) A = 2 0 −4 6 −4 3 3 (b) A = 0 0 1 3 0 0 1 3 6. Mark each of the following True or False. (a) Every n × n matrix is diagonalizable. (b) If an n × n matrix has n distinct real eigenvalues, then it is diagonalizable. (c) Every n × n real symmetric matrix is real diagonalizable. (d) An n × n matrix is diagonalizable if and only if it has n real eigenvalues. (e) An n × n matrix is diagonalizable if and only if the algebraic multiplicity of each of its eigenvalues equals the geometric multiplicity. (f) Every invertible matrix is diagonalizable. (g) Every triagular matrix is diagonalizable. (h) If A and B are similar matrices and A is diagonalizable, then B is also diagonalizable. (i) If an n × n matrix A is diagonalizable, there is a unique diagonal matrix D that is similar to A. (j) If A and B are similar square matrices then det (A) = det (B). 7. Find the polynomial in P2 whose coordinate vector relative to the ordered basis B = x + x2 , x − x2 , 1 + x is [3, 1, 2] 19 8. Let B be an ordered basis of R3 and let E for . If 3 CE,B = 4 −1 find the corrdinate vector vB . = ([1, 0, 0], [0, 1, 0] , [0, 0, 1]) denote the standard ordered basis 1 2 2 1 2 , v= 5 2 1 −1 9. Let V be a vector space with ordered bases B and B 0 . If 1 2 0 CB,B 0 = 0 1 −2 and −1 0 1 v = 3b1 − 2b2 + b3 find the coordinate vector vB 0 . 10. Let B = ([1, 1] , [1, 0]) and B 0 = ([0, 1] , [1, 1]). Find the change of coordinates matrix from B to B 0 and B 0 to B. Verify that they are inverses of each other.

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