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# MATH 225 midterm 2 MATH 286

UIUC

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This 2 page Study Guide was uploaded by Jigisha Sampat on Tuesday April 5, 2016. The Study Guide belongs to MATH 286 at University of Illinois at Urbana-Champaign taught by in Spring 2016. Since its upload, it has received 23 views. For similar materials see Math 286 in Mathematics (M) at University of Illinois at Urbana-Champaign.

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Great notes!!! Thanks so much for doing this...

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Date Created: 04/05/16

LINEARLY INDEPENDENT MATRIX A matrix A for which Ax=0 has only the trivial solution We say that the columns of A SPAN R^n when every vector b in R^n is a linear combination of A LINEARLY INDEPENDENT SET: A set is considered to be linearly independent if: It does not contain the zero vector A set of vector does not contain one vector as the multiple of the other BASIS SET A basis set is a set containing no unnecessary vectors. A set of independent vectors can be considered a basis of a set. WHAT IS A SUBSPACE? A subspace H of a Vector space V is defined as the subset H of V which has the following three properties: For each vector u and v in H, u+v are in H For each vector u and v in H, cu and cv are in H The zero vector of V is in H INVERTIBLE MATRIX THEOREM A in an invertible matrix The determinant of A is not equal to 0 A is row equivalent to I A has n pivot positions Ax=0 has only the trivial solutions The columns form a linearly independent set The equation Ax=b has at least one solution for each b in R^n The columns of A span R^n There is a matrix C such that CA= I There is a matrix D such that AD= I NULL SPACES A null space is a set of solutions to the homogenous equation Ax=0 Spanning set of Nul A is automatically linearly independent If Nul A is != {0} then the number of vectors in the spanning set is equal to the number of free variables. COLUMN SPACES A col space is a set of all the linear combinations of the columns of A BASIS OF NUL (A) AND COL (A) To find the basis of Nul(A) we must transform the matrix [A 0] to its equivalent row reduced echelon form [B 0]. Use this reduced row echelon form to find the parametric form of the general solution of Ax=0. These solutions form the basis of Nul (A). A basis of Col (A) is formed by the pivots columns of A. (use the pivot columns of the original matrix, not the pivot columns of the reduced echelon form of the matrix.) Dimensions of Nul(A) and Col (A) The dimensions of Col (A) are the number of non-zero rows in an augmented matrix. Col(A) + Nul(A) = Number of Column in A (augmented matrix) DIMENSION OF A VECTOR The dimension of a matric vector is the number of vectors in the basis of that set of vectors. EIGENVECTORS An Eigen vector of an nxn matric A is a non-zero vextor x such that Ax = kx, for some scalar k. EIGENVALUES The scalar k is known as the eigenvalue of the matrix if (A-kI)x=0 has a non-trivial solution. For triangular matrices, the eigenvalues is equal to the elements of the diagonal. EIGENSPACE The set of containing the solutions of (A-kI)x=0 is known as the Eigenspace.

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