- A The Transpose Of 3 0 2 1 4 Is 2 1 0 4 B The Map L R R Defined By L X 2x 3 Is A Linear Transformation C 1 (450.81 KiB) Viewed 33 times
(a) The transpose of 3 0 2 -1 4 is 2 -1 0 4 (b) The map L: R → R defined by L(x) = 2x + 3 is a linear transformation. (c) If a n x n matrix A is upper triangular, then Ax = b has a unique solution for any b of size n x 1. (d) Let A be a square matrix then rank(A) = rank(A²). (e) Consider any two different planes in a three dimensional space. If they intersect with each other, then the intersection is a one-dimensional vector space under conventional vector summation and scalar multiplication. (f) For a n x n matrix A, if Ax = b has only one solution for some vector b of size n × 1, then Ax = d has one and only one solution for any vector d of size n × 1. (g) The left null space of a m x n matrix A has dimension m - rank(A). (h) If U is the row echelon form of A then U and A have exactly the same row spaces. (i) The dimension of the vector space consisting of all real symmetric n x n matrices is (n + 1)n/2. (j) Any invertible square matrix can be written as a product of some elementary matrices.