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Lecture_14.docx-Lecture 14 o Finding Eigenvalues and

Math 20F A00, Linear Algebra, Winter 2012 Midterm Exam 2 Review Midterm2 will cover section 2.1-2.3, 3.1-3.3, 4.1-4.6, close-book, close-notes. The cheat sheet is not allowed. • Matrix operations : addition, scalar multiple, matrix multiplication (row-column rule for computing AB), transpose and inverse • The inverse of a matrix A . 1. If A is a 2 × 2 matrix • a b c d ‚ and ad - bc 6 = 0, use the formula in Thm 4 in section 2.2 (pg 103) to compute the inverse, i.e., A - 1 = 1 ad - bc • d - b - c a ‚ 2. If A is a 3 × 3 matrix, use the formula derived by Cramer’s rule, i.e., Thm 8 in section 3.3 (pg 179) 3. For larger matrix A , do row reduction to make [ A | I ] ∼ [ I | B ], then matrix B = A - 1 . • Determinant 1. For a 2 × 2 or 3 × 3 matrix, there is a trick to compute the determinant. 2. The determinant of a (upper or lower) triangle matrix is the product of the entries on the main diagonal. 3. Remember how elementary row operations change determinant, which can be used to compute the determinant of any matrix by transforming it to triangle matrix. 4. For large matrices, if it contains many “0”, then use co-factor expansion across one row or one column that contains most 0. Pay attention to the plus or minus sign as in a checkerboard pattern. 5. For a structured matrix, ﬁrst guess the determinants of a few smaller matrices and prove it’s true for any size by principle of induction. • Vector space : 1. A subspace is a subset of vector space, which contains zero vector and is closed under vector addition and multiplication by scalars. 2. Important subspaces and how to ﬁnd bases: for a m × n matrix A and B is row equivalent to A in echelon form. Null space is a subspace of R n . Computing the solution of A x = 0 in the parametric vector form will produce a spanning set for Nul A ; Column space is a subspace of R m . The pivot columns of matrix A form a basis for Col A ; Row space is a subspace of R n . The non-zero rows of B form a basis for Row A . 3. A basis for subspace H is a linearly independent set that spans H . Suppose dim H = n . If a set of vectors satisfy two out of (1) linearly independent, (2) spans H and (3) contains n vectors, then it is a basis for H 1 4. Coordinate mapping: is a one-to-one linear transformation from V onto R n . Its applica- tion: the vector space P n (polynomial of degree at most n) is isomorphic to R n +1 . The linear dependency in P n can be veriﬁed by matrix algebra. Show some polynomials form a basis and ﬁnd the coordinate vector relative to this basis. 5. Rank: for a m × n matrix A Rank A = dimCol A = dimRow A = Rank A T 6 min( m, n ) Rank A + dimNul A = n 2

MATH | 18 | Linear Algebra | study guides | Math 20F A00  Linear Algebra  

Quiz 2 Math 20F, W2012 Student ID: Name: Discussion Session: Let A =   2 5 7 5 - 2 - 3 6 3 4   , B =   1 - 2 0 1 3 0   ,C = • 0 2 - 1 1 ‚ , and D = • 1 0 1 1 ‚ . Compute CB T ,2 C - D , det( A ), det( C ). Is A invertible? If so, ﬁnd its inverse. Hint of computing A - 1 : You can do R3-R2 or 2R1-R2 to make ‘1’ appear in the ﬁrst column and then use this as the pivot element to eliminate the other non-zero entries. You won’t lose points if you demonstrate you know how to do it (even if the ﬁnal numbers are wrong). It is ok to use Cramer’s rule, which may appear easier to compute the matrix inversion.

MATH | 18 | Linear Algebra | assessment | Quiz 2 Math 20F  W2012 Student

Math 20F A00, Linear Algebra, Spring 2011 Practice Problems for Midterm Exam 2 1. Let A =   - 5 - 1 9 3 1 - 2 1 0 2   . Is A invertible? If so, ﬁnd its inverse. 2. Let A =   0 - 1 3 2 1 - 2 1 0 2   ,B =   - 5 1 0 3 1 0   ,C =   1 0 0 0 1 3 0 0 1   . Find out ( AB ) T and CB . 3. Find the inverse of the matrix A = • 1 - 1 1 - 3 ‚ . 4. Let A =   1 - 1 2 1 - 3 1 3 7 5   and b =   0 - 5 7   . Solve A x = b . by Cramer’s rule. 5. Find all the co-factors, the adjugate matrix, and the determinant of the matrix     1 1 2 - 1 0 2 - 4 1 - 1 0 2 0 - 1 0 2 0     . 6. Calculate the determinant of each of the following matrices:   5 0 - 1 1 - 3 - 2 0 5 3   ;     1 1 2 - 1 0 0 - 4 0 - 1 3 12 0 1 2 - 4 1     ;     1 3 3 - 4 0 1 2 5 2 5 4 - 3 - 3 - 7 - 5 2     . 7. Show that det   1 1 1 a b c a 2 b 2 c 2   =( a - b )( b - c )( c - a ). 8. Let A,B,C be three 2 × 2 matrices and O is a 2 × 2 zero matrix. Show that det • A O B C ‚ = (det A )(det C ). 9. Let A be a 3 × 3 matrix and assume det A = 2, the co-factors C 11 = 2, C 12 = - 4, C 13 = 6. What is the solution to A x = e 1 with e 1 = (1 , 0 , 0) T ? 10. Determine if H is a subspace of R 3 : (a) H consists of all the vectors in R 3 with the product of all the components equal to 0; (b) H consists of all the vectors in R 3 with the ﬁrst component equal to 1; (c) H consists of all the vectors in R 3 with the ﬁrst component equal to 0; (d) H consists of all the vectors   2 a - b 2 b - c 2 c - a   with a, b, c all real numbers. 11. Is the set of all 2 × 2 diagonal matrices a subspace of the space of all 2 × 2 matrices? 12. Determine if     - 4 - 8 6 - 5     is in the span by     3 8 5 2     ,     - 5 7 - 8 - 2     and     - 9 - 6 3 9     . 1 13. Determine if • 2 1 ‚ is in the column space and null space of • - 2 4 - 1 2 ‚ . 14. Let v 1 and v 2 be two vectors in a vector space. Show that any three vectors in Span { v 1 , v 2 } must be linearly dependent. 15. Work out Exercise problem 33 of Section 4.2 of the textbook. 16. Show that p 0 ( t ) = 1, p 1 ( t )= t , and p 2 ( t )=1+ t + t 2 form a basis of the vector space P 2 . Find the coordinates of p ( t )=2 - t +3 t 2 relative to this basis. 17. Show that p 0 ( t ) = 1, p 1 ( t )=1 - t , p 2 ( t )=2 - 4 t + t 2 and p 3 ( t )=6 - 18 t +9 t 2 - t 3 form a basis of the vector space P 3 . Find the coordinates of p ( t )=5+5 t - 2 t 2 relative to this basis. 18. Find a basis for the column space, the null space, and row space of   1 0 - 1 5 0 1 0 2 2 0 2 - 1   . What is the rank of A ? What is the dimension of Nul A ? 19. What is the dimension of R 4 , P 4 , the space of all 4 × 4 matrices? 20. If A is a 4 × 7 matrix, what is the largest possible rank of A . If A has four pivot columns, is Col A = R 4 ? What is the dimension of Nul A ? Is Nul A = R 3 ? Explain your answers. True or false: 1. Let T be a one-to-one linear transformation from R 4 to R 4 and A be the matrix associated with T . A is invertible. 2. det( A + B ) = det A + det B , det( - A )= - det A, and det( AB ) = (det A )(det B ); 3. det A T = det A and det A - 1 = (det A ) - 1 ; 4. A square matrix is invertible if and only if its determinant is nonzero. 5. Two row equivalent matrices have the same null space. 6. Two row equivalent matrices have the same column space. 7. Two row equivalent matrices have the same row space. 8. The row space of A is the column space of A T ; 9. If the null space of a square matrix A is { 0 } then A is invertible; 10. If u and v are two vectors in R 3 then the rank of the matrix uv T is always 0 or 1. 11. rank A + rank B = rank ( A + B ), rank ( AB ) = (rank A )(rank B ); 12. dim Col A + dim Nul A = number of columns of A. 13. If the columns of an n × n matrix A are linearly independent, then columns of A span R n . 14. If A is a 3 × 3 matrix and equation A x =   1 0 0   has a unique solution, then A is invertible. 15. Let W = { • x y ‚ : xy > 0 } . W is a subspace of R 2 . 16. The columns of an invertible n × n matrix form a basis for R n . 17. The dimension of the vector space P 4 is 4. 18. If V is a vector space and dim V = n then n + 1 vectors in V must be linearly dependent. 19. If there exists a set { v 1 , ··· , v p } that spans the vector space V , then dim V 6 p . 20. If there exists a linearly independent set { v 1 , ··· , v p } in V , then dim V 6 p . 21. If B is any echelon form of A , and if B has three nonzero rows, then the ﬁrst three rows of A form a basis for Row A . 22. If B is any echelon form of A , then the pivot columns of B form a basis for Col A . 23. If A is m × n and rank A = m , then the linear transformation x → A x is one-to-one. 24. If A is m × n and the linear transformation x → A x is onto, then rank A = m . 2

MATH | 18 | Linear Algebra | exam | Math 20F A00  Linear Algebra  

Lecture_03.docx-Lecture 03  Geometric Distribution o

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