Question

True or False 1. If u, v are vectors in R"and lu + v1l = ||||| + ||v||, then u and v are orthogonal. 2. If p locates a point on a line l in Rand if n # 0 is normal to l, then any other point x on I must satisfy n.x=n.p. 3. A binary vector is a vector with two components which are integers modulo 2. 4. The set of solution vectors to the linear system Ax=b is a vector space. 5. If A is invertible, then the only solution to Ax=b is A-'b. 6. If A is an m x n matrix, then Ax = 0 has nontrivial solution if and only if n>m. 7. If a row of a matrix A is multiplied by a scalar c, then the rank of the new matrix is the same as the rank of A. 8. Elementary row operations do not change the rank of a matrix. 9. The inverse of a square matrix can be computed using only elementary row operations. 10. The rank of the product of two matrices is always equal to the smaller of the ranks of the individual matrices. 11. If the matrix B is obtained by interchanging two rows of the matrix A, then det B =det A. 12. If E is an elementary matrix, then det E = +1. 13. Every system of n linear equation in n unknowns can be solved by Cramer's Rule.

Answer 1

$\small 1.\textbf{False.}\qquad 2.\textbf{True.}\qquad 3.\textbf{False.}\qquad 4.\textbf{False.}\qquad 5.\textbf{True.}\qquad 6.\textbf{False.}\\ 7.\textbf{False.}\qquad 8.\textbf{True.}\qquad 9.\textbf{True.}\qquad 10.\textbf{False.}\qquad 11.\textbf{False.}\qquad 12.\textbf{False.}\qquad 13.\textbf{False.}$