Part I. True-False. For each statement below, circle \"T\" if the statement is a

Question

Part I. True-False. For each statement below, circle "T" if the statement is always true, and "F" if the statement is sometimes or always false. [10 pts] 1) T F RREF(A) is unique. 2) T F (AB)T ATBT. 3) T F ?f A ìs a singular matrix, then 0 is an eigenvalue of A. 4) T F The solution set to a consistent equation "Ax- b" is a vector space. 5) T F Every square matrix has a basis of eigenvectors. 6) T F Adding a column of A to a column of A does not change det(A). F F F F If m x n A has rank n, then "Ax = b" has exactly one solution. 8) 9) 10) The columns of an mxn matrix A form a basis for Col(A). T T T (f(t)g(t))-F(s)G(s). If you know the solution to y'(t) p(t)y q(t), then you know the solution to y(t)+ p(t)y O.

Explanation / Answer

1). T As RREF of a Matrix Is Always Unique

2). F Use Shoes Socks Idea.

3). T Singular Matrix Is Always Not Invertible So 0 Must be its eigenvalue.

4). F Maybe solution set of equation have not zero vector.

5). F That depends on Field. Matrix has eigenvectors if and only if it has eigenvalues. If Eigenvalues are Complex but Field is Real so No Eigenvalue then No Eigenvector.

6). T We can add any times of any Column(Row) to any Column(Row)

7). F Unique solution is when Rank is n and matrix is of nxn.

8). T The column space of an mxn matrix A, written as Col A, is the linear combinations of the columns of A. If A = [a1,…,an], then Col A = Span { a1,…,an}

9). T Convolution Theorem

10). F From solution of 1st you can't find the solution of 2nd . you have to do separately to find the solution .You can find bu Integrating factor as a First order linear differential equation is an equation of the form y' + P(x)y = Q(x).   

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