(25 points) mark as true or false; no justification necessary. (a) if a and b are n × n-matrices such that ab has linearly inde pendent columns, then a has linearly independent columns. (b) there is a linear transformation t : r7 ? r3 with two- dimensional kernel. (c) there is a set of four vectors which span r5. (d) if the columns of a n x n-matrix a are linearly independent, then so are the rows of a. (e) the set of polynomials p(t) satisfying p(1) 5 is a subspace of p5. (f) the set of vectors of the form (2x-y+z, r+3z,2u-77 - y -z -1) is a subspace of r4 e sum of the dimensions of the row space and the null space of a equals the number of rows of a. (g) th (h) the non-pivot columns of a matrix are always linearly de (i) if h pen- ace of r3, then there is a 3 x 3-matrix with ce v of 3 x 4-matrices contains a spanning set dent. is a subsp h col a. of 13 vectors.

Consider the following geometry problems in 3-space enter t or f depending on whether the statement is true or false. (you must enter t or f -- true and false will not work.) f equation editorequation editor 1. a plane and a line either intersect or are parallel t equation editorequation editor 2. two planes parallel to a third plane are parallel f equation editorequation editor 3. two lines either intersect or are parallel f equation editorequation editor 4. two planes orthogonal to a third plane are parallel f equation editorequation editor 5. two planes orthogonal to a line are parallel f equation editorequation editor 6. two planes either intersect or are parallel f equation editorequation editor 7. two lines orthogonal to a third line are parallel t equation editorequation editor 8. two lines parallel to a third line are parallel t equation editorequation editor 9. two planes parallel to a line are parallel f equation editorequation editor 10. two lines parallel to a plane are parallel f equation editorequation editor 11. two lines orthogonal to a plane are parallel