Mathematics
Mathematics, 12.08.2019 19:30, smart1212

In a qr factorization, say a=qr (when a has linearly independent columns), the columns of q form an orthonormal basis for the column space of a.
true / false.

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Mathematics, 29.06.2019 03:30, jfrey7621
(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.
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Mathematics, 10.07.2019 02:20, 23ricorvan
1. label the following statements as true or false. a. an injective linear map t : u ? v maps linear independent vectors to linearly independent vectors. b. any set of linearly independent vectors vi,v2, ,vk n rn can be extended to a basis by adding n- kunit vectors c. any subspace of r" is the solution-space of a linear system ax-0. d. elementary row operations on an m x n-matrix a don't change the space generated by the columns of a e. elementary row operations on an m x n-matrix a don't change the space generated by the rows of a f. a linear system ax- 0 of m -equations in n-unknowns has always a nontrivial solution ifnnm g. if an m x n-matrix a has r-many linearly independent rows than it also must have n many linearly independent columns h. the determinant function det is a linear function on the vector space of n x n-matrices l. let a be an n × n-matrix. then det(cd) c det(a). j. for any n x n-matrix a one has that det(a"(a) (a" 1s the transpose ofa.)
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Mathematics, 10.07.2019 17:10, michello3
9. (30 points) mark as true or false; no justification necessary (a) if all the columns of a and b are pivot, then so are the columns of ab. (b) if for, v2, vs) is linearly independent, then so is ( (c) there is a 5 × 7 matrix whose columns span r5 with 2 pivot columns (d) if b is in the column space of a, then every solution of azb is a least squares solution. (e) any n x n symmetric matrix has n distinct real eigenvalues. (0) an orthogonal matrix is orthogonally diagonalizable. (g) if {ui, v2,u3} ? s an orthogonal basis for w c rs, then so is 1501,-27] ) to find the eigenvalues of a, reduce a to echelon form. 0) any eigenvalue of a is an eigenvalue of the transpose matrix (h) similar matrices have the same eigenvectors.
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