Linear Algebra Chapter 2

Linear Algebra Chapter 2 Theorem 1 - CORRECT ANSWER-A, B, and C are matrices of the same size, and r and s are scalars. A+B=B+A (A+B)+C=A+(B+C) A+0=A r(A+B)=rA+rB r(sA)=(rs)A matrix multiplication - CORRECT ANSWER-if A is an mxn matrix and if B is an nxp matrix with columns b_1,....b_p, then the product AB is the mxp matrix whose columns are Ab_1,...Ab_p. That is, AB = A[b1 ... bp] = [Ab1 .... Abp] Each column of AB is a linear combination of A using weights from the corresponding column of B - CORRECT ANSWER- Row-Column Rule for Computing AB - CORRECT ANSWER-If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of the corresponding entries from row i of A and column j of B. If (AB)_ij denoes the (i,j)-entry in AB, and if A is an mxn matrix then (AB)_ij = a_1i*b_1j+a_2i*b_2j+....a_in*b_jn row_i(AB) = row_i(A)*B - CORRECT ANSWER- Matrix Multiplication - CORRECT ANSWER-A is mxn and B and C have sizes fr which the given operations are definted. A(BC) = (AB)C A(B+C)=AB+AC (B+C)A=BA+CA r(AB)=(rA)B=A(rB) for any scalar r I_mA=A=AI_m Matrix Multiplication Warnings - CORRECT ANSWER-AB ! always = BA AB=AC doesn't mean B=C Theorem 3 - CORRECT ANSWER-Let A and B denote matrices whose sies are appropriate for the following sums and products. (A^T)^T = A (A+B)^T = A^T+B^T (RA)^T = r(A^T) (AB)^T = B^T*A^T A^-1 * A = I and A * A^-1 = I - CORRECT ANSWER- invertible matrix: nonsigular non-invertible matrix: singular - CORRECT ANSWER- ad-bc != 0 A is invertible - CORRECT ANSWER-inverse = 1/(ad-bc)[d -b] [-c a] Theorem 5 - CORRECT ANSWER-If A is an invertible nxn matrix then for each b in R^n, the equation Ax-b has the unique solution x=A^-1b Theorem 6 - CORRECT ANSWER-If A is invertible then A^-1 is invertible and (A^-1)^-1=A If A and B are nxn invertible matrices then so is AB and (AB)C^-1=(B^-1)(A^-1) if A is an invertible matrix then so is A^T and (A^T)^-1 = (A^-1)^T Theorem 7 - CORRECT ANSWER-An nxn matrix A is invertible iff A is row equivalent to In. If so the same sequience of row operations that reduces A to In also transforms In into AC^-1 IMT. If A is a square nxn matrix then all of the following are true, or false - CORRECT ANSWER-A is invertible A is row equivalent to In A has n pivot positions The equation Ax=0 has only the trivial solution The columns of A form a linearly independent set The linear transformation x-Ax is one-to-one The equation Ax=b has at least one (only one) solution for each b in R^n The columns of A span R^n The linear transformation x-Ax maps R^n onto R^n There is an nxn matrix C such that CA=I (reverse order too) A^T is an invertible matrix If A and B are square and AB=I then A=B^-1 and B=A^-1 - CORRECT ANSWER- Theorem 9 - CORRECT ANSWER-If T is an R^n to R^n linear transformation and let A be the standard matrix for T. T is invertible if A is an invertible matrix. If so the linear transformation S given by S(x) = A^-1x is the unique function satisfying S(T(x))=x T(S(x))=x S is the inverse of T or T^-1